Journal of Inequalities in Pure and Applied Mathematics
|
|
- Rodger Cooper
- 6 years ago
- Views:
Transcription
1 Journal of Inequalities in Pure and Applied Mathematics APPLICATIONS OF DIFFERENTIAL SUBORDINATION TO CERTAIN SUBCLASSES OF MEROMORPHICALLY MULTIVALENT FUNCTIONS H.M. SRIVASTAVA AND J. PATEL Department of Mathematics and Statistics University of Victoria Victoria, British Columbia V8W 3P4 Canada. Department of Mathematics Utkal University Vani Vihar, Bhubaneswar 7514 Orissa, India. volume 6, issue 3, article 88, 25. Received 18 April, 25; accepted 5 July, 25. Communicated by: Th.M. Rassias Abstract Home Page c 2 Victoria University ISSN electronic:
2 Abstract By making use of the principle of differential subordination, the authors investigate several inclusion relationships and other interesting properties of certain subclasses of meromorphically multivalent functions which are defined here by means of a linear operator. They also indicate relevant connections of the various results presented in this paper with those obtained in earlier works. 2 Mathematics Subject Classification: Primary 3C45; Secondary 3D3, 33C2. Key words: Meromorphic functions, Differential subordination, Hadamard product or convolution, Multivalent functions, Linear operator, Hypergeometric function. The present investigation was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP7353 and, in part, by the University Grants Commission of India under its DRS Financial Assistance Program. 1 Introduction and Definitions Preliminary Lemmas The Main Subordination s and The Associated Functional Inequalities References Page 2 of 31
3 1. Introduction and Definitions For any integer m > p, let Σ p,m denote the class of all meromorphic functions fz normalized by 1.1 fz = z p + a k z k p N := {1, 2, 3, }, k=m which are analytic and p-valent in the punctured unit disk U = {z : z C and < z < 1} = U \ {}. For convenience, we write Σ p, p+1 = Σ p. If fz and gz are analytic in U, we say that fz is subordinate to gz, written symbolically as follows: f g in U or fz gz z U, if there exists a Schwarz function wz, which by definition is analytic in U with w = and wz < 1 z U such that Indeed it is known that fz = g wz z U. Page 3 of 31 fz gz z U = f = g and fu gu.
4 In particular, if the function gz is univalent in U, we have the following equivalence cf., e.g., [5]; see also [6, p. 4]: fz gz z U f = g and fu gu. For functions fz Σ p,m, given by 1.1, and gz Σ p,m defined by 1.2 gz = z p + b k z k m > p; p N, k=m we define the Hadamard product or convolution of fz and gz by 1.3 f gz := z p + a k b k z k =: g fz k=m m > p; p N; z U. Following the recent work of Liu and Srivastava [3], for a function fz in the class Σ p,m, given by 1.1, we now define a linear operator D n by and in general 1.4 D 1 fz = z p + k=m D n fz = D D n 1 fz = z p + D fz = fz, z p+1 p + k + 1a k z k fz =, z p = p + k + 1 n a k z k k=m z p+1 D n 1 fz z p n N. Page 4 of 31
5 It is easily verified from 1.4 that 1.5 z D n fz = D n+1 fz p + 1D n fz f Σ p,m ; n N := N {}. The case m = of the linear operator D n was introduced recently by Liu and Srivastava [3], who investigated among other things several inclusion relationships involving various subclasses of meromorphically p-valent functions, which they defined by means of the linear operator D n see also [2]. A special case of the linear operator D n for p = 1 and m = was considered earlier by Uralegaddi and Somanatha [13]. Aouf and Hossen [1] also obtained several results involving the operator D n for m = and p N. Making use of the principle of differential subordination as well as the linear operator D n, we now introduce a subclass of the function class Σ p,m as follows. Definition. For fixed parameters A and B 1 B < A 1, we say that a function fz Σ p,m is in the class Σ n p,ma, B, if it satisfies the following subordination condition: 1.6 zp+1 D n fz p 1 + Az 1 + Bz n N ; z U. In view of the definition of differential subordination, 1.6 is equivalent to the following condition: z p+1 D n fz + p Bz p+1 D n fz + pa < 1 z U. Page 5 of 31
6 For convenience, we write 1 2αp, 1 = Σ n p,m α, Σ n p,m where Σ n p,mα denotes the class of functions in Σ p,m satisfying the following inequality: In particular, we have R z p+1 D n fz > α α < p; z U. Σ n p, A, B = R n,p A, B, where R n,p A, B is the function class introduced and studied by Liu and Srivastava [3]. The function class Hp; A, B, considered by Mogra [7], happens to be a further special case of the Liu-Srivastava class R n,p A, B when n =. In the present paper, we derive several inclusion relationships for the function class Σ n p,ma, B and investigate various other properties of functions belonging to the class Σ n p,ma, B, which we have defined here by means of the linear operator D n. These include for example some mapping properties involving the linear operator D n. Relevant connections of the results presented in this paper with those obtained in earlier works are also pointed out. Page 6 of 31
7 2. Preliminary Lemmas In proving our main results, we need each of the following lemmas. Lemma 1 Miller and Mocanu [5]; see also [6]. Let the function hz be analytic and convex univalent in U with h = 1. Suppose also that the function φz given by 2.1 φz = 1 + c p+m z p+m + c p+m+1 z p+m+1 + is analytic in U. If 2.2 φz + zφ z γ then φz ψz = hz γ p + m z γ z p+m and ψz is the best dominant of 2.2. Rγ ; γ ; z U, t γ p+m 1 htdt hz z U, With a view to stating a well-known result Lemma 2 below, we denote by Pγ the class of functions ϕz given by 2.3 ϕz = 1 + b 1 z + b 2 z 2 +, which are analytic in U and satisfy the following inequality: Page 7 of 31 R ϕz > γ γ < 1; z U.
8 Lemma 2 cf., e.g., Pashkouleva [9]. Let the function ϕz, given by 2.3, be in the class Pγ. Then R{ϕz} 2γ γ 1 + z Lemma 3 see [12]. For γ 1, γ 2 < 1, γ < 1; z U. Pγ 1 Pγ 2 Pγ 3 The result is the best possible. For real or complex numbers a, b, and c c / Z Gauss hypergeometric function is defined by γ3 := 1 21 γ 1 1 γ 2. := {, 1, 2, }, the 2F 1 a, b; c; z = 1 + ab c z aa + 1bb z2 1! cc + 1 2! +. We note that the above series converges absolutely for z U and hence represents an analytic function in U see, for details, [14, Chapter 14]. Each of the identities asserted by Lemma 4 below is well-known cf., e.g., [14, Chapter 14]. Lemma 4. For real or complex parameters a, b, and c c / Z, t b 1 1 t c b 1 1 zt a dt = ΓbΓc b Γc 2F 1 a, b; c; z Rc > Rb > ; Page 8 of 31
9 F 1 a, b; c; z = 1 z a z 2F 1 a, c b; c; ; z 1 2F 1 a, b; c; z = 2 F 1 a, b 1; c; z + az c 2 F 1 a + 1, b; c + 1; z; a + b + 1 2F 1 a, b; a + b + 1 ; 1 π Γ 2 =. 2 2 a + 1 b + 1 Γ Γ 2 2 We now recall a result due to Singh and Singh [11] as Lemma 5 below. Lemma 5. Let Φz be analytic in U with Φ = 1 and R Φz > 1 2 z U. Then, for any function F z analytic in U, Φ F U is contained in the convex hull of F U. Page 9 of 31
10 3. The Main Subordination s and The Associated Functional Inequalities Unless otherwise mentioned, we shall assume throughout the sequel that m is an integer greater than p, and that 1 B < A 1, λ >, n N, and p N. 1. Let the function fz defined by 1.1 satisfy the following subordination condition: Then 1 λzp+1 D n fz + λz p+1 D n+1 fz p 3.1 zp+1 D n fz p Qz 1 + Az 1 + Bz 1 + Az 1 + Bz z U, z U. where the function Qz given by A + 1 A B B 1 + Bz F 1 1, 1; + 1; Bz B λp+m 1+Bz Qz = A 1 + z B = λp+m+1 is the best dominant of 3.1. Furthermore, 3.2 R zp+1 D n fz > ρ z U, p Page 1 of 31
11 where A + 1 A B B 1 B F 1 1, 1; + 1; B B λp+m B 1 ρ = A 1 B =. λp+m+1 The inequality in 3.2 is the best possible. Proof. Consider the function φz defined by 3.3 φz = zp+1 D n fz p z U. Then φz is of the form 2.1 and is analytic in U. Applying the identity 1.5 in 3.3 and differentiating the resulting equation with respect to z, we get 1 λzp+1 D n fz + λz p+1 D n+1 fz p = φz + λ zφ z 1 + Az 1 + Bz Now, by using Lemma 1 for γ = 1/λ, we deduce that zp+1 D n fz Qz p { } 1 z = λp + m z 1/λp+m { } 1/λp+m t At 1 + Bt dt z U. Page 11 of 31
12 A + 1 A B B 1 + Bz F 1 1, 1; + 1; Bz B λp+m 1+Bz = A 1 + z B =, λp+m+1 by change of variables followed by the use of the identities 2.4, 2.5, and 2.6 with b = 1 and c = a + 1. This proves the assertion 3.1 of 1. Next, in order to prove the assertion 3.2 of 1, it suffices to show that { } 3.4 inf R Qz = Q 1. z <1 Indeed, for z r < 1, 1 + Az R 1 + Bz Upon setting 1 Ar 1 Br z r < 1. Gs, z = 1 + Asz and 1 + Bsz 1 dνs = λp + m s{1/λp+m} 1 ds s 1, which is a positive measure on the closed interval [, 1], we get Qz = 1 Gs, z dνs, Page 12 of 31
13 so that R Qz 1 1 Asr dνs = Q r z r < 1. 1 Bsr Letting r 1 in the above inequality, we obtain the assertion 3.2 of 1. Finally, the estimate in 3.2 is the best possible as the function Qz is the best dominant of 3.1. For λ = 1 and m =, 1 yields the following result which improves the corresponding work of Liu and Srivastava [3, 1]. Corollary 1. The following inclusion property holds true for the function class R n,p A, B: R n+1,p A, B R n,p 1 2ϱ, 1 R n,p A, B, where A + 1 A B B 1 B 1 2 F 1 1, 1; 1 + 1; B B p B 1 ϱ = 1 A B =. p+1 The result is the best possible. Putting A = 1 2α p, B = 1, λ = 1, n =, and m = p + 2 Page 13 of 31 in 1, we get Corollary 2 below.
14 Corollary 2. If fz Σ p, p+2 satisfies the following inequality: { R z p+1 p + 2f z + zf z} > α α < p; z U, then R z p+1 f z π > α + p α 2 1 The result is the best possible. z U. Remark 1. From Corollary 2, we note that, if fz Σ p, p+2 satisfies the following inequality: R z p+1 {p + 2f z + zf pπ 2 z} > z U, 4 π then This result is the best possible. R z p+1 f z > z U. The result asserted by Remark 1 above was also obtained by Pap [8]. 2. If fz Σ n p,mα α < p, then { 3.5 R z p+1 1 λ D n fz + λ D n+1 fz } > α z < R, where The result is the best possible. 1 R = 1 + λ2 p + m 2 p+m λp + m. Page 14 of 31
15 Proof. We begin by writing 3.6 z p+1 D n fz = α + p αuz z U. Then, clearly, uz is of the form 2.1, is analytic in U, and has a positive real part in U. Making use of the identity 1.5 in 3.6 and differentiating the resulting equation with respect to z, we observe that z 1 p+1 λ D n fz + λ D n+1 fz + α 3.7 p α Now, by applying the following estimate [4]: in 3.7, we get zu z R{uz} 2p + mrp+m 1 r 2p+m z = r < 1 = uz + λzu z. z 1 p+1 λ D n fz + λ D n+1 fz + α 3.8 R p α R uz 2λp + mrp+m 1. 1 r 2p+m It is easily seen that the right-hand side of 3.8 is positive, provided that r < R, where R is given as in 2. This proves the assertion 3.5 of 2. Page 15 of 31
16 In order to show that the bound R is the best possible, we consider the function fz Σ p,m defined by Noting that z p+1 D n fz = α + p α 1 + z p+m α < p; z U. 1 zp+m for z 1 p+1 λ D n fz + λ D n+1 fz + α p α we complete the proof of 2. = 1 z2p+m + 2λp + mz p+m 1 z 2p+m = iπ z = R exp, p + m Putting λ = 1 in 2, we deduce the following result. Corollary 3. If fz Σ n p,mα α < p, then fz Σ n+1 p,m α for z < R, where 1 p+m R = 1 + p + m2 p + m. The result is the best possible. Page 16 of 31
17 3. Let fz Σ n p,ma, B and let 3.9 F δ,p fz = δ z δ+p Then z 3.1 zp+1 D n F δ,p fz p t δ+p 1 ftdt δ > ; z U. Θz 1 + Az 1 + Bz z U, where the function Θz given by A + 1 A B B 1 + Bz 1 δ 2 F 1 1, 1; + 1; Bz B p+m 1+Bz Θz = 1 + Aδ z B = δ+p+m is the best dominant of 3.1. Furthermore, 3.11 R zp+1 D n F δ,p fz > κ z U, p where A + 1 A B B 1 B 1 δ 2 F 1 1, 1; + 1; B B p+m B 1 κ = 1 Aδ B =. δ+p+m The result is the best possible. Page 17 of 31
18 Proof. Setting 3.12 φz = zp+1 D n F δ,p fz p z U, we note that φz is of the form 2.1 and is analytic in U. Using the following operator identity: 3.13 z D n F δ,p fz = δ D n fz δ + p D n F δ,p fz in 3.12, and differentiating the resulting equation with respect to z, we find that zp+1 D n fz = φz + zφ z 1 + Az z U. p δ 1 + Bz Now the remaining part of 3 follows by employing the techniques that we used in proving 1 above. Setting m = in 3, we obtain the following result which improves the corresponding work of Liu and Srivastava [3, 2]. Corollary 4. If δ > and fz R n,p A, B, then F δ,p fz R n,p 1 2ξ, 1 R n,p A, B, where A + 1 A B B 1 B 1 2 F 1 1, 1; δ + 1; B B p B 1 ξ = 1 Aδ B =. δ+p The result is the best possible. Page 18 of 31
19 Remark 2. By observing that 3.14 z p+1 D n F δ,p fz = δ z δ Corollary 4 can be restated as follows. z f Σ p,m ; z U, t δ+p D n ft dt If δ > and fz R n,p A, B, then R δ z t δ+p D n ft dt > ξ z U, pz δ where ξ is given as in Corollary 4. In view of 3.14, 3 for A = 1 2α p, B = 1, and n = yields Corollary 5. If δ > and if fz Σ p,m satisfies the following inequality: R z p+1 f z > α α < p; z U, then R δz z δ t δ+p f tdt > α + p α The result is the best possible. δ [2F 1 1, 1; p + m + 1; 1 ] 1 2 z U. Page 19 of 31
20 4. Let fz Σ p,m. Suppose also that gz Σ p,m satisfies the following inequality: R z p D n gz > z U. If then where Proof. Letting R = D n fz D n gz 1 < 1 n N ; z U, R z D n fz > z < R D n, fz 9p + m2 + 4p2p + m 3p + m. 22p + m 3.15 wz = Dn fz D n gz 1 = κ p+mz p+m + κ p+m+1 z p+m+1 +, we note that wz is analytic in U, with w = and wz z p+m Then, by applying the familiar Schwarz lemma, we get wz = z p+m Ψz, z U. Page 2 of 31
21 where the function Ψz is analytic in U and Ψz 1 z U. Therefore, 3.15 leads us to 3.16 D n fz = D n gz 1 + z p+m Ψz z U. Making use of logarithmic differentiation in 3.16, we obtain 3.17 z D n fz D n fz = z D n gz D n gz + zp+m{ p + mψz + zψ z }. 1 + z p+m Ψz Setting φz = z p D n gz, we see that the function φz is of the form 2.1, is analytic in U, Rφz > z U, and so that we find from 3.17 that 3.18 R z D n fz z D n gz D n gz = zφ z φz p, D n fz p zφ z φz z p+m{ p + mψz + zψ z } 1 + z p+m Ψz z U. Page 21 of 31
22 Now, by using the following known estimates [1] see also [4]: φ z 2p + mrp+m 1 φz and 1 r 2p+m p + mψz + zψ z p + m 1 + z p+m Ψz z = r < 1 1 rp+m in 3.18, we obtain z D n fz R D n fz p 3p + mrp+m 2p + mr 2p+m 1 r 2p+m z = r < 1, which is certainly positive, provided that r < R, R being given as in Let 1 B j < A j 1 j = 1, 2. If each of the functions f j z Σ p satisfies the following subordination condition: λz p D n f j z + λz p D n+1 f j z 1 + A jz 1 + B j z then λz p D n Hz + λz p D n+1 Hz where Hz = D n f 1 f 2 z ηz 1 z j = 1, 2; z U, z U, Page 22 of 31
23 and η = 1 4A 1 B 1 A 2 B 2 1 B 1 1 B 2 The result is the best possible when B 1 = B 2 = 1. [ F 1 1, 1; 1 λ + 1; 1 ]. 2 Proof. Suppose that each of the functions f j z Σ p j = 1, 2 satisfies the condition Then, by letting 3.21 ϕ j z = 1 λz p D n f j z + λz p D n+1 f j z j = 1, 2, we have ϕ j z Pγ j γ j = 1 A j ; j = 1, 2. 1 B j By making use of the operator identity 1.5 in 3.21, we observe that D n f j z = 1 z λ z p 1/λ t 1/λ 1 ϕ j tdt j = 1, 2, which, in view of the definition of Hz given already with 3.2, yields 3.22 D n Hz = 1 z λ z p 1/λ t 1/λ 1 ϕ tdt, where, for convenience, 3.23 ϕ z = 1 λz p D n Hz + λz p D n+1 Hz = 1 z λ z 1/λ t 1/λ 1 ϕ 1 ϕ 2 tdt. Page 23 of 31
24 Since ϕ 1 z Pγ 1 and ϕ 2 z Pγ 2, it follows from Lemma 3 that 3.24 ϕ 1 ϕ 2 z Pγ 3 γ3 = 1 21 γ 1 1 γ 2. Now, by using 3.24 in 3.23 and then appealing to Lemma 2 and Lemma 4, we get R { ϕ z } = 1 λ 1 λ > 1 λ u 1/λ 1 R { ϕ 1 ϕ 2 } uzdu u 1/λ 1 2γ γ u z u 1/λ 1 2γ γ u = 1 4A 1 B 1 A 2 B 2 1 B 1 1 B 2 = 1 4A 1 B 1 A 2 B 2 1 B 1 1 B 2 = η z U. du du λ [ F 1 1, 1; 1 λ + 1; 1 2 u 1/λ u 1 du ] When B 1 = B 2 = 1, we consider the functions f j z Σ p j = 1, 2, which satisfy the hypothesis 3.19 of 5 and are defined by D n f j z = 1 z 1 + λ z 1/λ t 1/λ 1 Aj t dt j = 1, 2. 1 z Page 24 of 31
25 Thus it follows from 3.23 and Lemma 4 that ϕ z = 1 1 u 1/λ A A A 11 + A 2 du λ 1 uz = A A A A 2 1 z 1 2F 1 1, 1; 1 λ + 1; z z A A A 11 + A 2 2 F 1 1, 1; 1 λ + 1; 1 2 which evidently completes the proof of 5. By setting A j = 1 2α j, B j = 1 j = 1, 2, and n = as z 1, in 5, we obtain the following result which refines the work of Yang [15, 4]. Corollary 6. If the functions f j z Σ p j = 1, 2 satisfy the following inequality: 3.25 R 1 + λpz p f j z + λz p+1 f jz > α j α j < 1; j = 1, 2; z U, Page 25 of 31
26 then where R 1 + λpz p f 1 f 2 z + λz p+1 f 1 f 2 z > η z U, η = 1 41 α 1 1 α 2 [ F 1 1, 1; 1 λ + 1; 1 ]. 2 The result is the best possible. 6. If fz Σ p,m satisfies the following subordination condition: 1 λz p D n fz + λz p D n+1 fz 1 + Az 1 + Bz z U, then R z p D n fz 1/q > ρ 1/q q N; z U, where ρ is given as in 1. The result is the best possible. Proof. Defining the function φz by 3.26 φz = z p D n fz f Σ p,m ; z U, we see that the function φz is of the form 2.1 and is analytic in U. Using the identity 1.5 in 3.26 and differentiating the resulting equation with respect to z, we find that Page 26 of 31 1 λz p D n fz + λz p D n+1 fz = φz + λ zφ z 1 + Az 1 + Bz z U.
27 Now, by following the lines of proof of 1 mutatis mutandis, and using the elementary inequality: R w 1/q Rw 1/q Rw > ; q N, we arrive at the result asserted by 6. A = Upon setting [2F 1 1, 1; 1 λp + m + 1; 1 ] 1 [2 2 F 1 1, 1; 2 B = 1, n =, and q = 1 in 6, we deduce Corollary 7 below. Corollary 7. If fz Σ p,m satisfies the following inequality: 1 λp + m + 1; 1 ] 1, R 1 + λpz p fz + λz p+1 f z F 1 1, 1; + 1; 1 λp+m 2 > ] z U, 1 2 [2 2 F 1 1, 1; + 1; 1 λp+m 2 then The result is the best possible. R z p fz > 1 2 z U. Page 27 of 31
28 From Corollary 6 and 6 for m = p + 1, A = 1 2η, B = 1, and q = 1, we deduce the following result. Corollary 8. If the functions f j z Σ p j = 1, 2 satisfy the inequality 3.25, then R z p f 1 f 2 z > η + 1 η [2F 1 1, 1; 1 λ + 1; 1 ] 1 z U, 2 where η is given as in Corollary 6. The result is the best possible. 7. Let fz Σ n p,ma, B and let gz Σ p,m satisfy the following inequality: Then Proof. We have Since zp+1 D n f gz p and the function R z p gz > 1 2 z U. f gz Σ n p,ma, B. = zp+1 D n fz z p gz p R z p gz > Az 1 + Bz z U z U. Page 28 of 31
29 is convex univalent in U, it follows from 1.6 and Lemma 5 that f gz Σ n p,ma, B. This completes the proof of 7. In view of Corollary 7 and 7, we have Corollary 9 below. Corollary 9. If fz Σ n p,ma, B and the function gz Σ p,m satisfies the inequality 3.27, then f gz Σ n p,ma, B. Page 29 of 31
30 References [1] M.K. AOUF AND H.M. HOSSEN, New criteria for meromorphic p-valent starlike functions, Tsukuba J. Math., , [2] J.-L. LIU AND H.M. SRIVASTAVA, Classes of meromorphically multivalent functions associated with the generalized hypergeometric function, Math. Comput. Modelling, 39 24, [3] J.-L. LIU AND H.M. SRIVASTAVA, Subclasses of meromorphically multivalent functions associated with a certain linear operator, Math. Comput. Modelling, 39 24, [4] T.H. MacGREGOR, Radius of univalence of certain analytic functions, Proc. Amer. Math. Soc., , [5] S.S. MILLER AND P.T. MOCANU, Differential subordinations and univalent functions, Michigan Math. J., , [6] S.S. MILLER AND P.T. MOCANU, Differential Subordinations: Theory and Applications, Series on Monographs and Textbooks in Pure and Applied Mathematics, Vol. 225, Marcel Dekker, New York and Basel, 2. [7] M.L. MOGRA, Meromorphic multivalent functions with positive coefficients. I and II, Math. Japon., , 1 11 and [8] M. PAP, On certain subclasses of meromorphic m-valent close-to-convex functions, Pure Math. Appl., , Page 3 of 31
31 [9] D.Ž. PASHKOULEVA, The starlikeness and spiral-convexity of certain subclasses of analytic functions, in : H.M. Srivastava and S. Owa Editors, Current Topics in Analytic Function Theory, pp , World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, [1] J. PATEL, Radii of γ-spirallikeness of certain analytic functions, J. Math. Phys. Sci., , [11] R. SINGH AND S. SINGH, Convolution properties of a class of starlike functions, Proc. Amer. Math. Soc., , [12] J. STANKIEWICZ AND Z. STANKIEWICZ, Some applications of the Hadamard convolution in the theory of functions, Ann. Univ. Mariae Curie-Skłodowska Sect. A, , [13] B.A. URALEGADDI AND C. SOMANATHA, New criteria for meromorphic starlike univalent functions, Bull. Austral. Math. Soc., , [14] E.T. WHITTAKER AND G.N. WATSON, A Course on Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; With an Account of the Principal Transcendental Functions, Fourth Edition Reprinted, Cambridge University Press, Cambridge, [15] D.-G. YANG, Certain convolution operators for meromorphic functions, Southeast Asian Bull. Math., 25 21, Page 31 of 31
Majorization Properties for Subclass of Analytic p-valent Functions Defined by the Generalized Hypergeometric Function
Tamsui Oxford Journal of Information and Mathematical Sciences 284) 2012) 395-405 Aletheia University Majorization Properties for Subclass of Analytic p-valent Functions Defined by the Generalized Hypergeometric
More informationCoecient bounds for certain subclasses of analytic functions of complex order
Hacettepe Journal of Mathematics and Statistics Volume 45 (4) (2016), 1015 1022 Coecient bounds for certain subclasses of analytic functions of complex order Serap Bulut Abstract In this paper, we introduce
More informationDIFFERENTIAL SUBORDINATION RESULTS FOR NEW CLASSES OF THE FAMILY E(Φ, Ψ)
DIFFERENTIAL SUBORDINATION RESULTS FOR NEW CLASSES OF THE FAMILY E(Φ, Ψ) Received: 05 July, 2008 RABHA W. IBRAHIM AND MASLINA DARUS School of Mathematical Sciences Faculty of Science and Technology Universiti
More informationConvolution properties for subclasses of meromorphic univalent functions of complex order. Teodor Bulboacă, Mohamed K. Aouf, Rabha M.
Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Filomat 26:1 (2012), 153 163 DOI: 10.2298/FIL1201153B Convolution properties for subclasses of meromorphic
More informationSOME SUBCLASSES OF ANALYTIC FUNCTIONS DEFINED BY GENERALIZED DIFFERENTIAL OPERATOR. Maslina Darus and Imran Faisal
Acta Universitatis Apulensis ISSN: 1582-5329 No. 29/212 pp. 197-215 SOME SUBCLASSES OF ANALYTIC FUNCTIONS DEFINED BY GENERALIZED DIFFERENTIAL OPERATOR Maslina Darus and Imran Faisal Abstract. Let A denote
More informationSUBCLASSES OF P-VALENT STARLIKE FUNCTIONS DEFINED BY USING CERTAIN FRACTIONAL DERIVATIVE OPERATOR
Sutra: International Journal of Mathematical Science Education Technomathematics Research Foundation Vol. 4 No. 1, pp. 17-32, 2011 SUBCLASSES OF P-VALENT STARLIKE FUNCTIONS DEFINED BY USING CERTAIN FRACTIONAL
More informationOn a New Subclass of Meromorphically Multivalent Functions Defined by Linear Operator
International Mathematical Forum, Vol. 8, 213, no. 15, 713-726 HIKARI Ltd, www.m-hikari.com On a New Subclass of Meromorphically Multivalent Functions Defined by Linear Operator Ali Hussein Battor, Waggas
More informationCoefficient bounds for some subclasses of p-valently starlike functions
doi: 0.2478/v0062-02-0032-y ANNALES UNIVERSITATIS MARIAE CURIE-SKŁODOWSKA LUBLIN POLONIA VOL. LXVII, NO. 2, 203 SECTIO A 65 78 C. SELVARAJ, O. S. BABU G. MURUGUSUNDARAMOORTHY Coefficient bounds for some
More informationFUNCTIONS WITH NEGATIVE COEFFICIENTS
A NEW SUBCLASS OF k-uniformly CONVEX FUNCTIONS WITH NEGATIVE COEFFICIENTS H. M. SRIVASTAVA T. N. SHANMUGAM Department of Mathematics and Statistics University of Victoria British Columbia 1V8W 3P4, Canada
More informationA Certain Subclass of Multivalent Functions Involving Higher-Order Derivatives
Filomat 30:1 (2016), 113 124 DOI 10.2298/FIL1601113S Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat A Certain Subclass of Multivalent
More informationOn Certain Class of Meromorphically Multivalent Reciprocal Starlike Functions Associated with the Liu-Srivastava Operator Defined by Subordination
Filomat 8:9 014, 1965 198 DOI 10.98/FIL1409965M Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat On Certain Class of Meromorphically
More informationA New Subclasses of Meromorphic p-valent Functions with Positive Coefficient Defined by Fractional Calculus Operators
Volume 119 No. 15 2018, 2447-2461 ISSN: 1314-3395 (on-line version) url: http://www.acadpubl.eu/hub/ http://www.acadpubl.eu/hub/ A New Subclasses of Meromorphic p-valent Functions with Positive Coefficient
More informationThe Order of Starlikeness of New p-valent Meromorphic Functions
Int. Journal of Math. Analysis, Vol. 6, 2012, no. 27, 1329-1340 The Order of Starlikeness of New p-valent Meromorphic Functions Aabed Mohammed and Maslina Darus School of Mathematical Sciences Faculty
More informationSubclass of Meromorphic Functions with Positive Coefficients Defined by Frasin and Darus Operator
Int. J. Open Problems Complex Analysis, Vol. 8, No. 1, March 2016 ISSN 2074-2827; Copyright c ICSRS Publication, 2016 www.i-csrs.org Subclass of Meromorphic Functions with Positive Coefficients Defined
More informationInclusion and argument properties for certain subclasses of multivalent functions defined by the Dziok-Srivastava operator
Advances in Theoretical Alied Mathematics. ISSN 0973-4554 Volume 11, Number 4 016,. 361 37 Research India Publications htt://www.riublication.com/atam.htm Inclusion argument roerties for certain subclasses
More informationSome Geometric Properties of a Certain Subclass of Univalent Functions Defined by Differential Subordination Property
Gen. Math. Notes Vol. 20 No. 2 February 2014 pp. 79-94 SSN 2219-7184; Copyright CSRS Publication 2014 www.i-csrs.org Available free online at http://www.geman.in Some Geometric Properties of a Certain
More informationMeromorphic Starlike Functions with Alternating and Missing Coefficients 1
General Mathematics Vol. 14, No. 4 (2006), 113 126 Meromorphic Starlike Functions with Alternating and Missing Coefficients 1 M. Darus, S. B. Joshi and N. D. Sangle In memoriam of Associate Professor Ph.
More informationA NEW SUBCLASS OF MEROMORPHIC FUNCTION WITH POSITIVE COEFFICIENTS
Bulletin of Mathematical Analysis and Applications ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 3 Issue 32010), Pages 109-121. A NEW SUBCLASS OF MEROMORPHIC FUNCTION WITH POSITIVE COEFFICIENTS S.
More informationDifferential Sandwich Theorem for Multivalent Meromorphic Functions associated with the Liu-Srivastava Operator
KYUNGPOOK Math. J. 512011, 217-232 DOI 10.5666/KMJ.2011.51.2.217 Differential Sandwich Theorem for Multivalent Meromorhic Functions associated with the Liu-Srivastava Oerator Rosihan M. Ali, R. Chandrashekar
More informationJournal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied Mathematics INTEGRAL MEANS INEQUALITIES FOR FRACTIONAL DERIVATIVES OF SOME GENERAL SUBCLASSES OF ANALYTIC FUNCTIONS TADAYUKI SEKINE, KAZUYUKI TSURUMI, SHIGEYOSHI
More informationNEW SUBCLASS OF MULTIVALENT HYPERGEOMETRIC MEROMORPHIC FUNCTIONS
Kragujevac Journal of Mathematics Volume 42(1) (2018), Pages 83 95. NEW SUBCLASS OF MULTIVALENT HYPERGEOMETRIC MEROMORPHIC FUNCTIONS M. ALBEHBAH 1 AND M. DARUS 2 Abstract. In this aer, we introduce a new
More informationSOME CRITERIA FOR STRONGLY STARLIKE MULTIVALENT FUNCTIONS
Bulletin of the Institute of Mathematics Academia Sinica (New Series) Vol. 7 (01), No. 3, pp. 43-436 SOME CRITERIA FOR STRONGLY STARLIKE MULTIVALENT FUNCTIONS DINGGONG YANG 1 AND NENG XU,b 1 Department
More informationA NEW CLASS OF MEROMORPHIC FUNCTIONS RELATED TO CHO-KWON-SRIVASTAVA OPERATOR. F. Ghanim and M. Darus. 1. Introduction
MATEMATIQKI VESNIK 66, 1 14, 9 18 March 14 originalni nauqni rad research paper A NEW CLASS OF MEROMORPHIC FUNCTIONS RELATED TO CHO-KWON-SRIVASTAVA OPERATOR F. Ghanim and M. Darus Abstract. In the present
More informationSUBORDINATION RESULTS FOR CERTAIN SUBCLASSES OF UNIVALENT MEROMORPHIC FUNCTIONS
SUBORDINATION RESULTS FOR CERTAIN SUBCLASSES OF UNIVALENT MEROMORPHIC FUNCTIONS, P.G. Student, Department of Mathematics,Science College, Salahaddin University, Erbil, Region of Kurdistan, Abstract: In
More informationThe Relationships Between p valent Functions and Univalent Functions
DOI:.55/auom-25-5 An. Şt. Univ. Ovidius Constanţa Vol. 23(),25, 65 72 The Relationships Between p valent Functions and Univalent Functions Murat ÇAĞLAR Abstract In this paper, we obtain some sufficient
More informationOn Multivalent Functions Associated with Fixed Second Coefficient and the Principle of Subordination
International Journal of Mathematical Analysis Vol. 9, 015, no. 18, 883-895 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ijma.015.538 On Multivalent Functions Associated with Fixed Second Coefficient
More informationON CERTAIN CLASSES OF MULTIVALENT FUNCTIONS INVOLVING A GENERALIZED DIFFERENTIAL OPERATOR
Bull. Korean Math. Soc. 46 (2009), No. 5,. 905 915 DOI 10.4134/BKMS.2009.46.5.905 ON CERTAIN CLASSES OF MULTIVALENT FUNCTIONS INVOLVING A GENERALIZED DIFFERENTIAL OPERATOR Chellian Selvaraj and Kuathai
More informationMeromorphic parabolic starlike functions with a fixed point involving Srivastava-Attiya operator
Malaya J. Mat. 2(2)(2014) 91 102 Meromorphic parabolic starlike functions with a fixed point involving Srivastava-Attiya operator G. Murugusundaramoorthy a, and T. Janani b a,b School of Advanced Sciences,
More informationAN EXTENSION OF THE REGION OF VARIABILITY OF A SUBCLASS OF UNIVALENT FUNCTIONS
AN EXTENSION OF THE REGION OF VARIABILITY OF A SUBCLASS OF UNIVALENT FUNCTIONS SUKHWINDER SINGH SUSHMA GUPTA AND SUKHJIT SINGH Department of Applied Sciences Department of Mathematics B.B.S.B. Engineering
More informationRosihan M. Ali and V. Ravichandran 1. INTRODUCTION
TAIWANESE JOURNAL OF MATHEMATICS Vol. 14, No. 4, pp. 1479-1490, August 2010 This paper is available online at http://www.tjm.nsysu.edu.tw/ CLASSES OF MEROMORPHIC α-convex FUNCTIONS Rosihan M. Ali and V.
More informationON A DIFFERENTIAL SUBORDINATION AND SUPERORDINATION OF NEW CLASS OF MEROMORPHIC FUNCTIONS
LE MATEMATICHE Vol. LXIX 2014) Fasc. I, pp. 259 274 doi: 10.4418/2014.69.1.20 ON A DIFFERENTIAL SUBORDINATION AND SUPERORDINATION OF NEW CLASS OF MEROMORPHIC FUNCTIONS ALI MUHAMMAD - MARJAN SAEED In this
More informationSufficient conditions for certain subclasses of meromorphic p-valent functions
Bol. Soc. Paran. Mat. (3s.) v. 33 (015): 9 16. c SPM ISSN-175-1188 on line ISSN-0037871 in ress SPM: www.sm.uem.br/bsm doi:10.569/bsm.v33i.1919 Sufficient conditions for certain subclasses of meromorhic
More informationMULTIVALENTLY MEROMORPHIC FUNCTIONS ASSOCIATED WITH CONVOLUTION STRUCTURE. Kaliyapan Vijaya, Gangadharan Murugusundaramoorthy and Perumal Kathiravan
Acta Universitatis Apulensis ISSN: 1582-5329 No. 30/2012 pp. 247-263 MULTIVALENTLY MEROMORPHIC FUNCTIONS ASSOCIATED WITH CONVOLUTION STRUCTURE Kaliyapan Vijaya, Gangadharan Murugusundaramoorthy and Perumal
More informationA. Then p P( ) if and only if there exists w Ω such that p(z)= (z U). (1.4)
American International Journal of Research in Science, Technology, Engineering & Mathematics Available online at http://www.iasir.net ISSN (Print): 2328-3491, ISSN (Online): 2328-3580, ISSN (CD-ROM): 2328-3629
More informationDifferential subordination theorems for new classes of meromorphic multivalent Quasi-Convex functions and some applications
Int. J. Adv. Appl. Math. and Mech. 2(3) (2015) 126-133 (ISSN: 2347-2529) Journal homepage: www.ijaamm.com International Journal of Advances in Applied Mathematics and Mechanics Differential subordination
More informationAn Investigation on Minimal Surfaces of Multivalent Harmonic Functions 1
General Mathematics Vol. 19, No. 1 (2011), 99 107 An Investigation on Minimal Surfaces of Multivalent Harmonic Functions 1 Hakan Mete Taştan, Yaşar Polato glu Abstract The projection on the base plane
More informationFekete-Szegö Inequality for Certain Classes of Analytic Functions Associated with Srivastava-Attiya Integral Operator
Applied Mathematical Sciences, Vol. 9, 015, no. 68, 3357-3369 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ams.015.543 Fekete-Szegö Inequality for Certain Classes of Analytic Functions Associated
More informationDIFFERENTIAL SUBORDINATION AND SUPERORDINATION OF ANALYTIC FUNCTIONS DEFINED BY THE MULTIPLIER TRANSFORMATION
M athematical I nequalities & A pplications Volume 12, Number 1 2009), 123 139 DIFFERENTIAL SUBORDINATION AND SUPERORDINATION OF ANALYTIC FUNCTIONS DEFINED BY THE MULTIPLIER TRANSFORMATION ROSIHAN M. ALI,
More informationDifferential Operator of a Class of Meromorphic Univalent Functions With Negative Coefficients
Differential Operator of a Class of Meromorphic Univalent Functions With Negative Coefficients Waggas Galib Atshan and Ali Hamza Abada Department of Mathematics College of Computer Science and Mathematics
More informationOn a New Subclass of Salagean Type Analytic Functions
Int. Journal of Math. Analysis, Vol. 3, 2009, no. 14, 689-700 On a New Subclass of Salagean Type Analytic Functions K. K. Dixit a, V. Kumar b, A. L. Pathak c and P. Dixit b a Department of Mathematics,
More informationORDER. 1. INTRODUCTION Let A denote the class of functions of the form
SARAJEVO JOURNAL OF MATHEMATICS Vol.10 (22) (2014), 55 60 DOI: 10.5644/SJM.10.1.07 ON UNIFIED CLASS OF γ-spirallike FUNCTIONS OF COMPLEX ORDER T. M. SEOUDY ABSTRACT. In this paper, we obtain a necessary
More informationMajorization for Certain Classes of Meromorphic Functions Defined by Integral Operator
Int. J. Open Problems Complex Analysis, Vol. 5, No. 3, November, 2013 ISSN 2074-2827; Copyright c ICSRS Publication, 2013 www.i-csrs.org Majorization for Certain Classes of Meromorphic Functions Defined
More informationOn sandwich theorems for p-valent functions involving a new generalized differential operator
Stud. Univ. Babeş-Bolyai Math. 60(015), No. 3, 395 40 On sandwich theorems for p-valent functions involving a new generalized differential operator T. Al-Hawary, B.A. Frasin and M. Darus Abstract. A new
More informationSOME CLASSES OF MEROMORPHIC MULTIVALENT FUNCTIONS WITH POSITIVE COEFFICIENTS INVOLVING CERTAIN LINEAR OPERATOR
International Journal of Basic & Alied Sciences IJBAS-IJENS Vol:13 No:03 56 SOME CLASSES OF MEROMORPHIC MULTIVALENT FUNCTIONS WITH POSITIVE COEFFICIENTS INVOLVING CERTAIN LINEAR OPERATOR Abdul Rahman S
More informationSome Further Properties for Analytic Functions. with Varying Argument Defined by. Hadamard Products
International Mathematical Forum, Vol. 10, 2015, no. 2, 75-93 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2015.412205 Some Further Properties for Analytic Functions with Varying Argument
More informationSOME PROPERTIES OF A SUBCLASS OF ANALYTIC FUNCTIONS DEFINED BY A GENERALIZED SRIVASTAVA-ATTIYA OPERATOR. Nagat. M. Mustafa and Maslina Darus
FACTA UNIVERSITATIS NIŠ) Ser. Math. Inform. Vol. 27 No 3 2012), 309 320 SOME PROPERTIES OF A SUBCLASS OF ANALYTIC FUNCTIONS DEFINED BY A GENERALIZED SRIVASTAVA-ATTIYA OPERATOR Nagat. M. Mustafa and Maslina
More informationA NOTE ON UNIVALENT FUNCTIONS WITH FINITELY MANY COEFFICIENTS. Abstract
A NOTE ON UNIVALENT FUNCTIONS WITH FINITELY MANY COEFFICIENTS M. Darus, R.W. Ibrahim Abstract The main object of this article is to introduce sufficient conditions of univalency for a class of analytic
More informationOn a subclass of n-close to convex functions associated with some hyperbola
General Mathematics Vol. 13, No. 3 (2005), 23 30 On a subclass of n-close to convex functions associated with some hyperbola Mugur Acu Dedicated to Professor Dumitru Acu on his 60th anniversary Abstract
More informationConvex Functions and Functions with Bounded Turning
Tamsui Oxford Journal of Mathematical Sciences 26(2) (2010) 161-172 Aletheia University Convex Functions Functions with Bounded Turning Nikola Tuneski Faculty of Mechanical Engineering, Karpoš II bb, 1000
More informationSANDWICH-TYPE THEOREMS FOR A CLASS OF INTEGRAL OPERATORS ASSOCIATED WITH MEROMORPHIC FUNCTIONS
East Asian Mathematical Journal Vol. 28 (2012), No. 3, pp. 321 332 SANDWICH-TYPE THEOREMS FOR A CLASS OF INTEGRAL OPERATORS ASSOCIATED WITH MEROMORPHIC FUNCTIONS Nak Eun Cho Abstract. The purpose of the
More informationDIFFERENTIAL SUBORDINATION FOR MEROMORPHIC MULTIVALENT QUASI-CONVEX FUNCTIONS. Maslina Darus and Imran Faisal. 1. Introduction and preliminaries
FACTA UNIVERSITATIS (NIŠ) SER. MATH. INFORM. 26 (2011), 1 7 DIFFERENTIAL SUBORDINATION FOR MEROMORPHIC MULTIVALENT QUASI-CONVEX FUNCTIONS Maslina Darus and Imran Faisal Abstract. An attempt has been made
More informationResearch Article Properties of Certain Subclass of Multivalent Functions with Negative Coefficients
International Journal of Mathematics and Mathematical Sciences Volume 2012, Article ID 362312, 21 pages doi:10.1155/2012/362312 Research Article Properties of Certain Subclass of Multivalent Functions
More informationConvolution and Subordination Properties of Analytic Functions with Bounded Radius Rotations
International Mathematical Forum, Vol., 7, no. 4, 59-7 HIKARI Ltd, www.m-hikari.com https://doi.org/.988/imf.7.665 Convolution and Subordination Properties of Analytic Functions with Bounded Radius Rotations
More informationAbstract. For the Briot-Bouquet differential equations of the form given in [1] zu (z) u(z) = h(z),
J. Korean Math. Soc. 43 (2006), No. 2, pp. 311 322 STRONG DIFFERENTIAL SUBORDINATION AND APPLICATIONS TO UNIVALENCY CONDITIONS José A. Antonino Astract. For the Briot-Bouquet differential equations of
More informationSOME INCLUSION PROPERTIES OF STARLIKE AND CONVEX FUNCTIONS ASSOCIATED WITH HOHLOV OPERATOR. II
italian journal of pure and applied mathematics n. 37 2017 117 126 117 SOME INCLUSION PROPERTIES OF STARLIKE AND CONVEX FUNCTIONS ASSOCIATED WITH HOHLOV OPERATOR. II M. Kasthuri K. Vijaya 1 K. Uma School
More informationA NOVEL SUBCLASS OF UNIVALENT FUNCTIONS INVOLVING OPERATORS OF FRACTIONAL CALCULUS P.N. Kamble 1, M.G. Shrigan 2, H.M.
International Journal of Applied Mathematics Volume 30 No. 6 2017, 501-514 ISSN: 1311-1728 printed version; ISSN: 1314-8060 on-line version doi: http://dx.doi.org/10.12732/ijam.v30i6.4 A NOVEL SUBCLASS
More informationSOME APPLICATIONS OF SALAGEAN INTEGRAL OPERATOR. Let A denote the class of functions of the form: f(z) = z + a k z k (1.1)
STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume LV, Number 1, March 2010 SOME APPLICATIONS OF SALAGEAN INTEGRAL OPERATOR Abstract. In this paper we introduce and study some new subclasses of starlike, convex,
More informationSome basic properties of certain subclasses of meromorphically starlike functions
Wang et al. Journal of Inequalities Applications 2014, 2014:29 R E S E A R C H Open Access Some basic properties of certain subclasses of meromorphically starlike functions Zhi-Gang Wang 1*, HM Srivastava
More informationarxiv: v1 [math.cv] 19 Jul 2012
arxiv:1207.4529v1 [math.cv] 19 Jul 2012 On the Radius Constants for Classes of Analytic Functions 1 ROSIHAN M. ALI, 2 NAVEEN KUMAR JAIN AND 3 V. RAVICHANDRAN 1,3 School of Mathematical Sciences, Universiti
More informationSubclasses of Analytic Functions. Involving the Hurwitz-Lerch Zeta Function
International Mathematical Forum, Vol. 6, 211, no. 52, 2573-2586 Suclasses of Analytic Functions Involving the Hurwitz-Lerch Zeta Function Shigeyoshi Owa Department of Mathematics Kinki University Higashi-Osaka,
More informationCertain properties of a new subclass of close-to-convex functions
Arab J Math (212) 1:39 317 DOI 1.17/s465-12-29-y RESEARCH ARTICLE Pranay Goswami Serap Bulut Teodor Bulboacă Certain properties of a new subclass of close-to-convex functions Received: 18 November 211
More informationOn a subclass of certain p-valent starlike functions with negative coefficients 1
General Mathematics Vol. 18, No. 2 (2010), 95 119 On a subclass of certain p-valent starlike functions with negative coefficients 1 S. M. Khairnar, N. H. More Abstract We introduce the subclass T Ω (n,
More informationDifferential Subordination and Superordination for Multivalent Functions Involving a Generalized Differential Operator
Differential Subordination Superordination for Multivalent Functions Involving a Generalized Differential Operator Waggas Galib Atshan, Najah Ali Jiben Al-Ziadi Departent of Matheatics, College of Coputer
More informationAn ordinary differentail operator and its applications to certain classes of multivalently meromorphic functions
Bulletin of Mathematical analysis and Applications ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 1, Issue 2, (2009), Pages 17-22 An ordinary differentail operator and its applications to certain
More informationCertain classes of p-valent analytic functions with negative coefficients and (λ, p)-starlike with respect to certain points
BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Number 379), 2015, Pages 35 49 ISSN 1024 7696 Certain classes of p-valent analytic functions with negative coefficients and λ, p)-starlike
More informationSOME RESULTS ASSOCIATED WITH FRACTIONAL CALCULUS OPERATORS INVOLVING APPELL HYPERGEOMETRIC FUNCTION
Volume 29), Issue, Article 4, 7 pp. SOME RESULTS ASSOCIATED WITH FRACTIONAL CALCULUS OPERATORS INVOLVING APPELL HYPERGEOMETRIC FUNCTION R. K. RAINA / GANPATI VIHAR, OPPOSITE SECTOR 5 UDAIPUR 332, RAJASTHAN,
More informationRosihan M. Ali, M. Hussain Khan, V. Ravichandran, and K. G. Subramanian. Let A(p, m) be the class of all p-valent analytic functions f(z) = z p +
Bull Korean Math Soc 43 2006, No 1, pp 179 188 A CLASS OF MULTIVALENT FUNCTIONS WITH NEGATIVE COEFFICIENTS DEFINED BY CONVOLUTION Rosihan M Ali, M Hussain Khan, V Ravichandran, and K G Subramanian Abstract
More informationInclusion relationships for certain classes of analytic functions involving the Choi-Saigo-Srivastava operator
Cho Yoon Journal of Inequalities Applications 2013, 2013:83 R E S E A R C H Open Access Inclusion relationships for certain classes of analytic functions involving the Choi-Saigo-Srivastava operator Nak
More informationCoefficient Bounds for a Certain Class of Analytic and Bi-Univalent Functions
Filomat 9:8 (015), 1839 1845 DOI 10.98/FIL1508839S Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Coefficient Bounds for a Certain
More informationON SOME LENGTH PROBLEMS FOR ANALYTIC FUNCTIONS
Nunokawa, M. and Sokół, J. Osaka J. Math. 5 (4), 695 77 ON SOME LENGTH PROBLEMS FOR ANALYTIC FUNCTIONS MAMORU NUNOKAWA and JANUSZ SOKÓŁ (Received December 6, ) Let A be the class of functions Abstract
More informationA SUBORDINATION THEOREM WITH APPLICATIONS TO ANALYTIC FUNCTIONS
Bulletin of Mathematical Analysis and Applications ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 3 Issue 3(2011, Pages 1-8. A SUBORDINATION THEOREM WITH APPLICATIONS TO ANALYTIC FUNCTIONS (COMMUNICATED
More informationAn Application of Wilf's Subordinating Factor Sequence on Certain Subclasses of Analytic Functions
International Journal of Applied Engineering Research ISS 0973-4562 Volume 3 umber 6 (208 pp. 2494-2500 An Application of Wilf's Subordinating Factor Sequence on Certain Subclasses of Analytic Functions
More informationOn a subclass of analytic functions involving harmonic means
DOI: 10.1515/auom-2015-0018 An. Şt. Univ. Ovidius Constanţa Vol. 231),2015, 267 275 On a subclass of analytic functions involving harmonic means Andreea-Elena Tudor Dorina Răducanu Abstract In the present
More informationINTEGRAL MEANS OF UNIVALENT SOLUTION FOR FRACTIONAL EQUATION IN COMPLEX PLANE. Rabha W. Ibrahim and Maslina Darus
Acta Universitatis Apulensis ISSN: 1582-5329 No. 23/21 pp. 39-47 INTEGRAL MEANS OF UNIVALENT SOLUTION FOR FRACTIONAL EQUATION IN COMPLEX PLANE Rabha W. Ibrahim and Maslina Darus Abstract. Integral means
More informationConvolutions of Certain Analytic Functions
Proceedings of the ICM2010 Satellite Conference International Workshop on Harmonic and Quasiconformal Mappings (HQM2010) Editors: D. Minda, S. Ponnusamy, and N. Shanmugalingam J. Analysis Volume 18 (2010),
More informationProperties of Starlike Functions with Respect to Conjugate Points
Int. J. Contemp. Math. Sciences, Vol. 5, 1, no. 38, 1855-1859 Properties of Starlike Functions with Respect to Conjugate Points Suci Aida Fitri Mad Dahhar and Aini Janteng School of Science and Technology,
More informationResearch Article A Study of Cho-Kwon-Srivastava Operator with Applications to Generalized Hypergeometric Functions
International Mathematics and Mathematical Sciences, Article ID 374821, 6 pages http://dx.doi.org/10.1155/2014/374821 Research Article A Study of Cho-Kwon-Srivastava Operator with Applications to Generalized
More informationCERTAIN DIFFERENTIAL SUBORDINATIONS USING A GENERALIZED SĂLĂGEAN OPERATOR AND RUSCHEWEYH OPERATOR. Alina Alb Lupaş
CERTAIN DIFFERENTIAL SUBORDINATIONS USING A GENERALIZED SĂLĂGEAN OPERATOR AND RUSCHEWEYH OPERATOR Alina Alb Lupaş Dedicated to Prof. H.M. Srivastava for the 70th anniversary Abstract In the present paper
More informationAn Integrodifferential Operator for Meromorphic Functions Associated with the Hurwitz-Lerch Zeta Function
Filomat 30:7 (2016), 2045 2057 DOI 10.2298/FIL1607045A Pulished y Faculty of Sciences Mathematics, University of Niš, Seria Availale at: http://www.pmf.ni.ac.rs/filomat An Integrodifferential Operator
More informationTHE FEKETE-SZEGÖ COEFFICIENT FUNCTIONAL FOR TRANSFORMS OF ANALYTIC FUNCTIONS. Communicated by Mohammad Sal Moslehian. 1.
Bulletin of the Iranian Mathematical Society Vol. 35 No. (009 ), pp 119-14. THE FEKETE-SZEGÖ COEFFICIENT FUNCTIONAL FOR TRANSFORMS OF ANALYTIC FUNCTIONS R.M. ALI, S.K. LEE, V. RAVICHANDRAN AND S. SUPRAMANIAM
More informationa n z n, z U.. (1) f(z) = z + n=2 n=2 a nz n and g(z) = z + (a 1n...a mn )z n,, z U. n=2 a(a + 1)b(b + 1) z 2 + c(c + 1) 2! +...
ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.13(2012) No.2,pp.153-157 Integral Operator Defined by Convolution Product of Hypergeometric Functions Maslina Darus,
More informationDifferential Subordination and Superordination Results for Certain Subclasses of Analytic Functions by the Technique of Admissible Functions
Filomat 28:10 (2014), 2009 2026 DOI 10.2298/FIL1410009J Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: htt://www.mf.ni.ac.rs/filomat Differential Subordination
More informationA Certain Subclass of Multivalent Analytic Functions Defined by Fractional Calculus Operator
British Journal of Mathematics & Comuter Science 4(3): 43-45 4 SCIENCEDOMAIN international www.sciencedomain.org A Certain Subclass of Multivalent Analytic Functions Defined by Fractional Calculus Oerator
More informationTHE HYPERBOLIC METRIC OF A RECTANGLE
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 26, 2001, 401 407 THE HYPERBOLIC METRIC OF A RECTANGLE A. F. Beardon University of Cambridge, DPMMS, Centre for Mathematical Sciences Wilberforce
More informationTwo Points-Distortion Theorems for Multivalued Starlike Functions
Int. Journal of Math. Analysis, Vol. 2, 2008, no. 17, 799-806 Two Points-Distortion Theorems for Multivalued Starlike Functions Yaşar Polato glu Department of Mathematics and Computer Science TC İstanbul
More informationON CERTAIN CLASSES OF UNIVALENT MEROMORPHIC FUNCTIONS ASSOCIATED WITH INTEGRAL OPERATORS
TWMS J. App. Eng. Math. V.4, No.1, 214, pp. 45-49. ON CERTAIN CLASSES OF UNIVALENT MEROMORPHIC FUNCTIONS ASSOCIATED WITH INTEGRAL OPERATORS F. GHANIM 1 Abstract. This paper illustrates how some inclusion
More informationDIFFERENTIAL SUBORDINATION ASSOCIATED WITH NEW GENERALIZED DERIVATIVE OPERATOR
ROMAI J., v.12, no.1(2016), 77 89 DIFFERENTIAL SUBORDINATION ASSOCIATED WITH NEW GENERALIZED DERIVATIVE OPERATOR Anessa Oshah, Maslina Darus School of Mathematical Sciences, Faculty of Science and Technology,
More informationUniformly convex functions II
ANNALES POLONICI MATHEMATICI LVIII. (199 Uniformly convex functions II by Wancang Ma and David Minda (Cincinnati, Ohio Abstract. Recently, A. W. Goodman introduced the class UCV of normalized uniformly
More informationarxiv: v1 [math.cv] 12 Apr 2014
GEOMETRIC PROPERTIES OF BASIC HYPERGEOMETRIC FUNCTIONS SARITA AGRAWAL AND SWADESH SAHOO arxiv:44.327v [math.cv] 2 Apr 24 Abstract. In this paper we consider basic hypergeometric functions introduced by
More informationOn certain subclasses of analytic functions
Stud. Univ. Babeş-Bolyai Math. 58(013), No. 1, 9 14 On certain subclasses of analytic functions Imran Faisal, Zahid Shareef and Maslina Darus Abstract. In the present paper, we introduce and study certain
More informationResearch Article New Classes of Analytic Functions Involving Generalized Noor Integral Operator
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 390435, 14 pages doi:10.1155/2008/390435 Research Article New Classes of Analytic Functions Involving Generalized
More informationResearch Article A New Class of Meromorphically Analytic Functions with Applications to the Generalized Hypergeometric Functions
Abstract and Applied Analysis Volume 20, Article ID 59405, 0 pages doi:0.55/20/59405 Research Article A New Class of Meromorphically Analytic Functions with Applications to the Generalized Hypergeometric
More informationHarmonic multivalent meromorphic functions. defined by an integral operator
Journal of Applied Mathematics & Bioinformatics, vol.2, no.3, 2012, 99-114 ISSN: 1792-6602 (print), 1792-6939 (online) Scienpress Ltd, 2012 Harmonic multivalent meromorphic functions defined by an integral
More informationResearch Article A Study on Becker s Univalence Criteria
Abstract and Applied Analysis Volume 20, Article ID 75975, 3 pages doi:0.55/20/75975 Research Article A Study on Becker s Univalence Criteria Maslina Darus and Imran Faisal School of Mathematical Sciences,
More informationOn the improvement of Mocanu s conditions
Nunokawa et al. Journal of Inequalities Applications 13, 13:46 http://www.journalofinequalitiesapplications.com/content/13/1/46 R E S E A R C H Open Access On the improvement of Mocanu s conditions M Nunokawa
More informationOn subclasses of n-p-valent prestarlike functions of order beta and type gamma Author(s): [ 1 ] [ 1 ] [ 1 ] Address Abstract KeyWords
On subclasses of n-p-valent prestarlike functions of order beta and type gamma Elrifai, EA (Elrifai, E. A.) [ 1 ] ; Darwish, HE (Darwish, H. E.) [ 1 ] ; Ahmed, AR (Ahmed, A. R.) [ 1 ] E-mail Address: Rifai@mans.edu.eg;
More informationResearch Article Subordination and Superordination on Schwarzian Derivatives
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 008, Article ID 7138, 18 pages doi:10.1155/008/7138 Research Article Subordination and Superordination on Schwarzian Derivatives
More informationSUBCLASS OF HARMONIC STARLIKE FUNCTIONS ASSOCIATED WITH SALAGEAN DERIVATIVE
LE MATEMATICHE Vol. LXIX (2014) Fasc. II, pp. 147 158 doi: 10.4418/2014.69.2.13 SUBCLASS OF HARMONIC STARLIKE FUNCTIONS ASSOCIATED WITH SALAGEAN DERIVATIVE H. E. DARWISH - A. Y. LASHIN - S. M. SOILEH The
More informationSUBORDINATION AND SUPERORDINATION FOR FUNCTIONS BASED ON DZIOK-SRIVASTAVA LINEAR OPERATOR
Bulletin of Mathematical Analysis and Applications ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 2, Issue 3(21), Pages 15-26. SUBORDINATION AND SUPERORDINATION FOR FUNCTIONS BASED ON DZIOK-SRIVASTAVA
More informationA Note on Coefficient Inequalities for (j, i)-symmetrical Functions with Conic Regions
BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 2303-4874 ISSN (o) 2303-4955 wwwimviblorg /JOURNALS / BULLETIN Vol 6(2016) 77-87 Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA
More information