TAIWANESE JOURNAL OF MATHEMATICS Vol. 18, No. 1, pp. 39-51, February 14 DOI: 1.1165/tjm.18.14.986 This paper is available online at http://journal.taiwanmathsoc.org.tw GENERALIZATIONS OF STRONGLY STARLIKE FUNCTIONS Jacek Dziok Abstract. By using functions of bounded variation we generalize the class of strongly starlike functions and related classes. The main object is to obtain characterizations and inclusion properties of these classes of functions. 1. INTRODUCTION Let A denote the class of functions which are analytic in U := {z C : z < 1} and let A p p N := {, 1,,...} denote the class of functions f Aof the form 1 fz =z p + n=p+1 a n z n z U. Let a, δ C, a < 1, α<p, <β 1, k, p N, ϕ A p. A function f A p is said to be in the class Sβ of multivalent strongly starlike function of order β if Arg zf z pfz <βπ z U. We denote by M k the class of real-valued functions m of bounded variation on [, π] which satisfy the conditions π dm t =, π dm t k. It is clear that M is the class of nondecreasing functions on [, π] satisfying or π equivalently dm t =. Received February, 13, accepted May 16, 13. Communicated by Hari M. Srivastava. 1 Mathematics Subject Classification: 3C45, 3C5, 3C55. Key words and phrases: Analytic functions, Bounded variation, Bounded boundary rotation, Strongly starlike functions. 39
4 Jacek Dziok Let P k a, β denote the class of functions q A for which there exists m M k such that 3 q z = 1 π 1+1 a ze it β dm t z U. 1 ze it Here and throughout we assume that all powers denote principal determinations. Moreover, let us denote { } P k a :=P k a, 1, Pk a, β:= q A : q 1/β P k a. In particular, P := P is the well-known class of Carathéodory functions. The classes P k := P k, P k ρ ρ<1 were investigated by Paatero [19] see also Pinchuk [3] and Padmanabhan and Parvatham [1], respectively. We note that where P β :=P β = f S β zf z pfz P β, { q A : Arg q z <β π }. Now, we generalize the class of strongly starlike functions. We denote by M k a, β; δ, ϕ the class of functions f A p such that J δ,ϕ fz := δ 1+ z ϕ f z p ϕ f +1 δ z ϕ f z z p ϕ fz P k a, β, where denote the Hadamard product or convolution. Moreover, let us denote M a, β; δ, ϕ:=m a, β; δ, ϕ, W k a, β; ϕ:=m k a, β;,ϕ, W a, β; ϕ:=w a, β; ϕ, W k a, β:=w k a, β; z p / 1 z, Sp ϕ, a :=W a, 1; ϕ. We see that S β = W,β and 4 f W k a, β; ϕ ϕ f W k a, β. Let a =a 1,a, β =β 1,β. We say that a function f A p belongs to the class CW a, k β ; δ, ϕ, if there exists a function g W k a,β ; ϕ such that δ 1+ z ϕ f z p ϕ g +1 δ z ϕ f z z p ϕ gz P k a 1,β 1. These classes generalize well-known classes of functions, which were defined in ealier works, see for example [1-1] and [14-6]. We note that
Generalizations of Strongly Starlike Functions 41 M k a, β; α, ϕ is related to the class of functions with the bounded Mocanu variation defined by Coonce and Ziegler [4] and intensively investigated by Noor et al. [15-18] z V k := W k, 1; is the well-known class of functions of bounded boundary rotation for details, see, [, 7, 14, 1 z 1]. z Sp α :=Sp p z 1 z,α/p, Sp c α :=Sp p p +1 p z p 1 z,α/p are the classes of multivalent starlike functions of order α and multivalent convex functions of order α, respectively. R p α :=Sp z p / 1 z p α,α/p α <p will be called the class of multivalent prestarlike functions of order α. In particular, R α :=R 1 α is the well-know class of prestarlike functions of order α introduced by Ruscheweyh [4]. z CC := CW k, 1; is the well-known class of close-to-convex functions. 1 z The main object of the paper is to obtain some characterizations and inclusion properties for the defined classes of functions. Some applications of the main results are also considered.. CHARACTERIZATION THEOREMS Let us define { k B k a, β := 4 + 1 k q 1 4 1 } q : q 1,q h a,β, where 1+1 a z β 5 h a,β z :=, h a := h a,1 z U. 1 z From the result of Hallenbeck and MacGregor [13], pp. 5 we have the following lemma. Lemma 1. q h a,β if and only if there exists m M such that π 1+1 a ze it β dm t z U. q z = 1 1 ze it
4 Jacek Dziok Theorem 1. B λ a, β B k a, β λ<k. Proof. Let q B λ a, β. Then there exist q 1,q h a,β such that q = λ 4 + 1 q1 λ 4 1 q or k q = 4 + 1 k q 1 4 1 q q = k λ k q 1 + λ k q. Since q h a,β,wehaveq B k a, β. h a,β Theorem. The class B k a, β is convex. Proof. Let q, r B k a, β,α [, 1], μ:= k 4 + 1. Then there exist q j,r j j =1, such that It follows that q = μq 1 1 μ q,r= μr 1 1 μ r. αq +1 α r = μ αq 1 +1 α r 1 1 μαq +1 α r. Since αq j +1 α r j h a,β j =1,, we conclude that αq+1 α r B k a, β. Hence, the class B k a, β is convex. Theorem 3. P k a, β =B k a, β. Proof. Let q P k a, β. Thenq satisfy 3 for some m M k. If m M, then by Lemma 1 and Theorem 1 we have q P h P k h. Let now m M k M. Since m is the function with bounded variation, by the Jordan theorem there exist real-valued functions μ 1,μ which are nondecreasing and nonconstant on [, π] such that 6 m = μ 1 μ, Thus, putting π dm t = π dμ 1 t+ π dμ t. α j = μ j π μ j, m j := 1 μ j α j j =1, we get m 1,m M and 7 m = α 1 m 1 α m. Combining 6 and 7 we obtain
Generalizations of Strongly Starlike Functions 43 and so α 1 α = α 1 = π dm t =, α 1 +α = λ 4 + 1 λ,α = 4 1 λ = π π dm t k, dm t k. Therefore, by 3 and 7 we obtain λ q = 4 + 1 λ q 1 4 1 q, where q j z = 1 π 1+1 a ze it β 1 ze it dm j t z U,j=1,. Thus, by Lemma 1 and Theorem 1 we have q B λ a, β B k a, β. Conversely, let q B k a, β. Then there exist q 1,q h a,β such that q is of the form k q = 4 + 1 k q 1 4 1 q. Thus, by Lemma 1 there exist m 1,m M such that q is of the form 3 with k m = 4 + 1 k m 1 4 1 m. Since π π dm t = dm t k 4 + 1 π dm 1 k 4 + 1 π dm 1 + we have m M k and consequently q P k a, β. k 4 1 π k 4 1 π Lemma. [7]. Letq A. Then q P k a if and only if π re it Rq a dt kπ <r<1. 1 a dm =, dm = k,
44 Jacek Dziok From Lemma we have the following corollary. 8 Corollary 1. Let q A.Thenq P k a, β if and only if π Rq1/β re it a dt kπ <r<1. 1 a 3. THE MAIN INCLUSION RELATIONSHIPS From now on we make the assumptions: δ 1 and 9 Rh a,β z >α z U. Then we have 1 W k a, β S p α. Let Φ p b, c denote the multivalent incomplete hipergeometric function defined by 11 Φ p b, cz :=z p F 1b, 1; c; z= n=p Lemma 3. [11]. Let h K,q A and λ>. If then q h. qz+λ zq z qz h z, Lemma 4. [5]. Let f R p α, g Sp α. Then f hg U co {hu}, f g where co {hu} denotes the closed convex hull of hu. Lemma 5. [5]. Letp N. If either b n p c n p z n z U. 1 R[b] R[c], I[b] =I[c] and 1 p +1 b c α<p or 13 <b c and p c α<p, then Φ p b, c R p α.
Generalizations of Strongly Starlike Functions 45 Theorem 4. If ψ R p α, then 14 W k a, β; ϕ W k a, β; ψ ϕ. Proof. that Let f W k a, β; ϕ. Thus, by Theorem 3 there exist q 1,q h a,β such z ϕ f z k p ϕ fz = 4 + 1 k q 1 z 4 1 q z z U. Moreover, H = ϕ f W k a, β Sp α. convolution, we get Thus, applying the properties of 15 z [ψ ϕ f] z p [ψ ϕ f]z = k 4 + 1 ψ z [q1 z Hz] ψ z Hz k 4 1 ψ z [q z Hz] ψ z Hz z U. By Lemma 4 we conclude that F j z := ψ z [q j z Hz] ψ z Hz co {q j U} h a,β U z U,j=1,. Therefore, F j h a,β and by 15 we have f W k a, β; ψ ϕ, which proves the theorem. Theorem 5. Let ψ R p α, δ 1. Then 16 M k a, β; δ, ϕ W k a, β; ϕ M k a, β; δ, ψ ϕ. Proof. Let f M k a, β; δ, ϕ W k a, β; ϕ. Then, applying Theorem 4, we obtain f W k a, β; ψ ϕ. Thus, we have F 1 z := z [ψ ϕ f] z p [ψ ϕ f]z,f z := z ϕ f z p ϕ fz P k a, β. Since the class P k a, β is convex by Theorem, we conclude that 1 δ F 1 +δf P k a, β. Hence, f M k a, β; δ, ψ ϕ and, in consequence, we get 16. Lemma 6. If γ δ, then M a, β; δ, ϕ Ma, β; γ,ϕ. Proof. Let f Ma, β; δ, ϕ and let q z := z ϕ f z p ϕ fz z U.
46 Jacek Dziok Then, we obtain q z+δ zq z q z = J δ,ϕ fz z U. Since J δ,ϕ f h a,β, we have q h a,β by Lemma 3. Moreover, J γ,ϕ f = γ δ J δ,ϕ f+ δ γ q. δ Because h a,β is convex and univalent in U, then we obtain J γϕ f h a,β or equivalently f Ma, β; γ,ϕ. From Theorem 5 and Lemma 6 we have the following corollary. Corollary. Let ψ R p α, δ 1. Then M a, β; δ, ϕ Ma, β; δ, ψ ϕ. Theorem 6. If ψ R p α, then 17 CW a, k β ; δ, ϕ CW a, k β ; δ, ψ ϕ. Proof. Let f CW a, k β ; δ, ϕ. Then there exist g W k a,β ; ϕ and q 1,q h a1,β 1 such that z ϕ f z k p ϕ gz = 4 + 1 k q 1 z 4 1 q z z U and F = ϕ g W k a,β Sp α. Thus, applying the properties of convolution, we get z [ψ ϕ f] z k 18 p [ψ ϕ g]z = 4 + 1 ψ q1 F ψ F z k 4 1 ψ q F z z U. ψ F By Lemma 4 we conclude that F j z := ψ q jf ψ F z co {q j U} h a1,β1 U z U, j =1,. Therefore, F j h a1,β 1 and by 18 we have f CW k a, β ; δ, ψ ϕ. Combining Theorems 4-6 with Lemma 5 we obtain the following theorem. Theorem 7. If either 1 or 13, then W k a, β; ϕ W k a, β;φ p b, c ϕ, M a, β; δ, ϕ Ma, β; δ, Φ p b, c ϕ, CW a, k β ; δ, ϕ CW a, k β ; δ, Φp b, c ϕ.
Generalizations of Strongly Starlike Functions 47 Since Φ p b, c Φ p c, b ϕ = ϕ, by Theorem 7 we obtain the next result. Theorem 8. If either 1 or 13, then W k a, β;φ p c, b ϕ W k a, β; ϕ, M a, β; δ, Φ p c, b ϕ Ma, β; δ, ϕ, CW a, k β ; δ, Φp c, b ϕ CW a, k β ; δ, ϕ. Let us define the linear operators J λ : A p A p, 19 J λ fz :=λ zf z +1 λ f z, z U, Rλ >. p Since J λ f =Φ p p λ +1, p λ f, putting b = p λ, c = p λ +1in Theorem 8, we have the following theorem. Theorem 9. If p R [ p λ] α<p,then W k a, β; J λ ϕ W k a, β; ϕ, M a, β; δ, J λ ϕ Ma, β; δ, ϕ, CW a, k β ; δ, Jλ ϕ CW a, k β ; δ, ϕ. In particular, for λ =1we get the following theorem. Theorem 1. If α<p,then W k a, β; zϕ z W k a, β; ϕ, M a, β; δ, zϕ z Ma, β; δ, ϕ, CW a, k β ; δ, zϕ z CW a, k β ; δ, ϕ. 4. APPLICATIONS TO CLASSES DEFINED BY LINEAR OPERATORS For real numbers λ, t λ > p, we define the function Ψa 1,b 1,tz:=z p qf s a 1,..., a q ; b 1,..., b s ; z f λ,t z z U, where q F s a 1,..., a q ; b 1,..., b s ; z is the generalized hypergeometric function and It is easy to verify that f λ,t z = n=p n + λ t z n z U. p + λ 1 bψb +1,c,t=zΨ b, c, t+b pψb, c, t,
48 Jacek Dziok 3 4 bψb, c, t=zψ b, c +1,t+b pψb, c +1,t, p + λψb, c, t +1=zΨ b, c, t+λψb, c, t, Ψb, c, t=φ p b, d Ψd, c, t. where Φ p b, d is defined by 11. Corresponding to the function Ψb, c, t we consider the following classes of functions: By using the linear operator V k a, β; b, c, t:=w k a, β;ψb, c, t, CV k a, β ; b, c, t:=cw k a, β ; δ, Ψb, c, t. 5 Θ p [b, c, t] f =Ψb, c, t f f A p we can define the class V k a, β; b, c, t alternatively in the following way: f V k a, β; b, c, t b Θ p [b +1,c,t] fz + p b P k a, β. Θ p [b, c, t] fz Corollary 3. If p R[b] α<p, m N, then 6 V k a, β; b + m, c, t V k a, β; b, c, t, 7 CV k a, β; b + m, c, t CV k a, β; b, c, t. Proof. It is clear that it is sufficient to prove the corollary for m =1. Let J λ and Ψb,c, t be defined by 19 and, respectively. Then, by 1 we have Ψb+1,c, t=jp Ψb, c, t. Hence, by using Theorem 9 we conclude that b W k a, β;ψb +1,c,t W k a, β;ψb, c, t, CW a, k β ; δ, Ψb +1,c,t CW a, k β ; δ, Ψb, c, t. This clearly forces the inclusion relations 6 and 7 for m =1. Analogously to Corollary 3, we prove the following corollary. Corollary 4. Let m N. If p R[c] α<p,then If R [λ] α<p,then V k a, β; b, c, t V k a,β; b,c + m, t, CV k a, β ; b, c, t CV k a, β ; b, c + m, t. V k a, β; b, c, t + m V k a, β; b, c, t, CV k a, β ; b, c, t + m CV k a, β ; b, c, t.
Generalizations of Strongly Starlike Functions 49 It is natural to ask about the inclusion relations in Corollaries 3 and 4 when m is positive real. Using Theorems 4 and 6, we shall give a partial answer to this question. Corollary 5. If the multivalent incomplete hipergeometric function Φ p b, d defined by 11 belongs to the class R p α, then V k a, β; d, c, t V k a, β;b,c, t, CV a, k β ;d, c, t CV a, 8 k β ; b, c,t, V k a, β; c, b, t V k a,β;c, d,t, CV a, k β ;c, b, t CV a, 9 k β ; c, d, t. Proof. Let us put ψ =Φ p b, d, ϕ=ψd, c, t. Then, by and Theorems 4 and 6 we obtain W k a, β;ψd, c, t W k a, β;ψb, c, t, CW k a, β ; δ, Ψd, c, t CW k a, β ; δ, Ψb, c, t. Thus, we get the inclusion relations 8. Analogously, we prove the inclusions 9. Combining Corollary 5 with Lemma 5, we obtain the following result. Corollary 6. If either 1 or 13, then the inclusion relations 8 and 9 hold true. The linear operator Θ p [b, c, t] defined by 5 includes as its special cases other linear operators of geometric function theory which were considered in earlier works. In particular, we can mention here the Dziok-Srivastava operator, the Hohlov operator, the Carlson-Shaffer operator, the Ruscheweyh derivative operator, the generalized Bernardi-Libera-Livingston operator, the fractional derivative operator, and so on for the precise relationships, see, Dziok and Srivastava [1], pp. 3-4. Moreover, the linear operator Θ p [b, c, t] includes also the Sălăgean operator, the Noor operator, the Choi-Saigo-Srivastava operator, the Kim-Srivastava operator, and others for the precise relationships, see, Cho et al. [3]. By using these linear operators we can consider several subclasses of the classes V k a, β; b, c, t and CV k a, β ; b, c, t, see for example [1-1, 1,,, 6]. Also, the obtained results generalize several results obtained in these classes of functions. REFERENCES 1. M. K. Aouf, Some inclusion relationships associated with Dziok Srivastava operator, Appl. Math. Comput., 16 1, 431-437.. S. Bhargava and R. S. Nanjunda, Convexity of a class of functions related to classes of starlike functions and functions with boundary rotation, Ann. Polon. Math., 493 1989, 9-35.
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