Research Article Some Inclusion Relationships of Certain Subclasses of p-valent Functions Associated with a Family of Integral Operators

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1 ISRN Mathematical Analysis Volume 2013, Article ID , 8 pages Research Article Some Inclusion Relationships of Certain Subclasses of p-valent Functions Associated with a Family of Integral Operators M. K. Aouf, 1 R. M. El-Ashwah, 2 and Ahmed M. Abd-Eltawab 3 1 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt 2 Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt 3 Department of Mathematics, Faculty of Science, Fayoum University, Fayoum 63514, Egypt Correspondence should be addressed to Ahmed M. Abd-Eltawab; ams03@fayoum.edu.eg Received 16 June 2013; Accepted 24 August 2013 Academic Editors: G. Gripenberg and B. Wang Copyright 2013 M. K. Aouf et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. By making use of the new integral operator, we introduce and investigate several new subclasses of p-valent starlike, p-valent convex, p-valent close-to-convex, and p-valent quasi-convex functions. In particular, we establish some inclusion relationships associated with the aforementioned integral operators. Some of the results established in this paper would provide extensions of those given in earlier works. 1. Introduction Let A(p) denotetheclassoffunctionsoftheform f (z) =z p + a k+p z k+p (p N = 1, 2, 3,...), (1) which are analytic and p-valent in the unit disc U=z:z C,and z < 1 and let A(1) = A. Afunctionf A(p)is said to be in the class S p (λ) of p-valent starlike functions of order λ in U if and only if Re ( zf (z) )>λ (z U; 0 λ<p). (2) f (z) The class S p (λ) was introduced by Patil and Thakare [1]. Owa [2]introducedtheclassK p (λ) of p-valent convex of order λ in U if and only if Re (1 + zf (z) f )>λ (z U; 0 λ<p). (3) (z) It is easy to observe from (2)and(3)that f (z) K p (λ) zf (z) p S p (λ). (4) We denote by S p = S p (0) and K p = K p (0) where S p and K p are the classes of p-valently starlike functions and pvalently convex functions, respectively, (see Goodman [3]). For a function f A(p),wesaythatf C p if there exists a function g S p (λ) such that Re ( zf (z) ) >η (z U; 0 λ,η<p). (5) Functions in the class C p are called p-valent close-toconvex functions of order η and type λ. TheclassC p was studied by Aouf [4] andtheclassc 1 was studied by Libera [5]. Noor [6, 7] introducedandstudiedtheclassesc p and C 1 as follows. Afunctionf A(p)is said to be in the class C p of quasi-convex functions of order η and type λ if there exists a function g K p (λ) such that Re (zf (z)) g (z) >η (z U; 0 λ, η<p). (6)

2 2 ISRN Mathematical Analysis It follows from (5)and(6)that f (z) C p zf (z) C p p. (7) For functions f A(p)given by (1) andg A(p)given by =z p + b k+p z k+p (p N), (8) the Hadamard product (or convolution) of f and g is given by (f g) (z) =z p + a k+p b k+p z k+p =(g f)(z). (9) For the function f A(p),weintroducedtheoperator : A(p) A(p) as follows: (α γ + 1) f (z) =(p+α+β γ) p+β 1 z β z (1 t 0 z )α γ t β 1 f (t) dt = Γ(p+α+β γ+1) 1 Γ(p+β)Γ(α γ+1) z β z (1 t 0 z )α γ t β 1 f (t) dt =z p + Γ(p+α+β γ+1) Γ(p+β) Γ(β+p+k) [ Γ(α+β+p+k γ+1) ]a k+pz k+p (β > p; α γ 1; γ > 0; p N; z U). (10) From (10), it is easy to verify that where the operator Q α was introduced and studied by Liu and Owa [8], and Q α β,1 = Qα β,wheretheoperatorqα β was introduced and studied by Jung et al. [9]; (ii) For α=γ=1and β=c, R 1,1 c,p f (z) =J c,p (z) = p+c z c =z p + z t c 1 f (t) dt 0 ( c+p c+k+p )a k+pz k+p (c > p; p N; z U), (13) where J c,p (z) is the familiar integral operator, which was defined by Cho and Kim [10]. The operator J c,1 (z) = J c (z) was introduced by Bernardi [11] and we note that J 1,1 (z) = J(z) was introduced and studied by Libera [12] andlivingston [13]. The main object of this paper is to investigate the various inclusion relationships for each of the following subclasses of thenormalizedanalyticfunctionclassa(p) which are defined here by means of the operator given by (10). Definition 2. In conjunction with (2)and(10), S α,β,γ,p (λ) =f:f A(p),Rα,γ f S p (λ), 0 λ<p,p N. Definition 3. In conjunction with (3)and(10), (14) =(α+β+p γ+1) f (z) (α+β γ+1) f (z). (11) Remark 1. Consider (i) For γ=1, R α,1 f (z) =Qα f (z) =( p+α+β 1 p+β 1 ) α z z (1 t β z )α 1 t β 1 f (t) dt = Γ(p+α+β) z 1 Γ(p+β)Γ(α) z (1 t β z )α 1 t β 1 f (t) dt =z p + Γ(p+α+β) Γ(p+β) 0 [ Γ(β+p+k) Γ(α+β+p+k) ]a k+pz k+p (β > p; α 0; p N; z U), 0 (12) K α,β,γ,p (λ) =f:f A(p), f K p (λ), 0 λ<p,p N. Definition 4. In conjunction with (5)and(10), C α,β,γ,p = f : f A (p), f C p, 0 λ,η<p,p N. Definition 5. In conjunction with (6)and(10), C α,β,γ,p = f : f A (p), Rα,γ f C p, 0 λ,η<p,p N. (15) (16) (17)

3 ISRN Mathematical Analysis 3 Remark 6. Consider (I) For γ=1, in the above definitions, we have S α,β,1,p (λ) =S α, (λ) =f:f A(p),Q α f S p (λ), K α,β,1,p (λ) =K α, (λ) 0 λ<p,p N, =f:f A(p),Q α f K p (λ), (III) For γ=p=1,intheabovedefinitions,wehave S α,β,1,1 (λ) =S α,β (λ) K α,β,1,1 (λ) =K α,β (λ) C α,β,1,1 = C α,β =f:f A,Q α β f S (λ),0 λ<1, =f:f A,Q α βf K(λ),0 λ<1, =f:f A,Q α β f C(η,λ),0 λ,η<1, 0 λ<p,p N, C α,β,1,p = C α, =f:f A(p),Q α f C p, (18) C α,β,1,1 = C α,β =f:f A,Q α β f C, 0 λ, η < 1, (20) 0 λ,η<p,p N, C α,β,1,p = C α, =f:f A(p),Q α f C p, 0 λ,η<p,p N. (II) For α=γ=1and β=c(c> p),intheabove definitions, we have S 1,c,1,p (λ) =S c,p (λ) K 1,c,1,p (λ) =K c,p (λ) =f:f A(p),J c,p f (z) S p (λ), 0 λ<p,p N, =f:f A(p),J c,p f (z) K p (λ), where the classes S α,β (λ), K α,β(λ), C α,β, andc α,β were introduced and studied by Gao et al. [14]. In order to establish our main results, we need the followinglemmaduetomillerandmocanu[15]. Lemma 7 (see [15]). Let Θ be a complex-valued function such that Θ:D C, D C C (C is the complex plane), (21) and let u=u 1 +iu 2, V = V 1 +iv 2.SupposethatΘ(u, V) satisfies the following conditions: Let (i) Θ(u, V) is continuous in D; (ii) (1, 0) D and ReΘ(1, 0) > 0; (iii) ReΘ(iu 2, V 1 ) 0 for all (iu 2, V 1 ) Dsuch that V 1 (1/2)(1 + u 2 2 ). q (z) =1+q 1 z+q 2 z 2 + (22) C 1,c,1,p = C c,p C 1,c,1,p = C c,p 0 λ<p,p N, =f:f A(p),J c,p f (z) C p, 0 λ,η<p,p N, =f:f A(p),J c,p f (z) C p, 0 λ,η<p,p N. (19) be analytic in U such that (q(z), z q (z)) D for all z U.If then Re Θ (q (z),z q (z))>0 (z U), (23) 2. The Main Results Re q (z) >0 (z U). (24) In this section, we give several inclusion relationships for analytic function classes, which are associated with the integral operator. Unless otherwise mentioned, we assume throughout this paper that β> p, α γ 1, γ>0, p N, and z U.

4 4 ISRN Mathematical Analysis Theorem 8. Let 0 λ<p.then Proof. Let f S α,β,γ,p (λ) and set S α,β,γ,p (λ) S α+1,β,γ,p (λ). (25) f(z)) f (z) λ = (p λ) q (z), (26) whereq(z) is given by (22).By usingidentity (11), we obtain f (z) λ+(p λ)q(z) +(α+β γ+1) =. (27) f (z) α+β+p γ+1 Differentiating (27) logarithmicallywithrespecttoz, we obtain z( f (z) = f (z) + = λ + (p λ) q (z) + (p λ) zq (z) (p λ) q (z) +λ+α+β γ+1 (p λ) zq (z) (p λ) q (z) +λ+α+β γ+1. (28) We now choose u=q(z)=u 1 +iu 2 and V =zq (z) = V 1 +iv 2, and define the function Θ by (p λ) V Θ (u, V) =(p λ)u+ (p λ)u+λ+α+β γ+1. (29) Then, clearly Θ(u,V) satisfies the following conditions: (i) Θ(u, V) is continuous in D=(C \(λ+α+β γ+ 1)/(λ p)) C; (ii) (1, 0) D and ReΘ(1, 0) = p λ > 0; (iii) for all (iu 2, V 1 ) Dsuch that V 1 (1/2)(1 + u 2 2 ) we have (p λ) V Re Θ (iu 2, V 1 ) = Re 1 (p λ) iu 2 +λ+α+β γ+1 (p λ) (λ + α + β γ + 1) V = 1 (p λ) 2 u2 2 +(λ+α+β γ+1)2 (p λ) (1 + u2 2 )(λ+α+β γ+1) 2[(p λ) 2 u 2 2 +(λ+α+β γ+1)2 ] <0, (30) which shows that the function Θ satisfies the hypotheses of Lemma 7. Consequently, we easily obtain the inclusion relationship (25). Theorem 9. Let 0 λ<p, p N.Then K α,β,γ,p (λ) K α+1,β,γ,p (λ). (31) Proof. Let f K α,β,γ,p (λ).then,fromdefinition 3,wehave f K p (λ), (0 λ<p; p N). (32) Furthermore, in view of the relationship (4), we find that that is, that z p (Rα,γ S p (λ), (33) (z) (zf p ) S p (λ). (34) Thus, by using Definition 2 and Theorem 8,wehave zf (z) p which implies that S p (α,β,γ,λ) S p (α+1,β,γ,λ), (35) K α,β,γ,p (λ) K α+1,β,γ,p (λ). (36) The proof of Theorem 9 is thus completed. Theorem 10. Let 0 λ, η<p.then C α,β,γ,p C α+1,β,γ,p. (37) Proof. Let f C α,β,γ,p. Then there exists a function Ψ S p (λ) such that Re We put z( Ψ (z) so that we have g S α,β,γ,p We next put (λ), Re >η (0 λ,η<p,z U). (38) =Ψ(z), (39) z( >η (z U). (40) =η+(p η)q(z), (41)

5 ISRN Mathematical Analysis 5 where q(z) is given by (22). Thus, by using identity (11), we obtain z( = (zf (z)) =(z[ (zf (z))] =( +(α+β γ+1) (zf (z))) (z[ ] ( +(α+β γ+1) 1 ) z[ (zf (z))] +(α+β γ+1) Rα+1,γ (zf (z)) ) z[ ] +(α+β γ+1)) 1. (42) Since g S α,β,γ,p (λ), thenfromtheorem 8 we have g S α+1,β,γ,p (λ),sothatwecanput where Then z( =( z[ ] = λ + (p λ) G (z), (43) G (z) =g 1 (x, y) + ig 2 (x, y), Re (G (z)) =g 1 (x, y) > 0 (z U). z[ (zf (z))] (44) +(α+β γ+1)(η+(p η)q(z))) ((p λ)g(z) +λ+α+β γ+1) 1. (45) We thus find from (41)that = [η+(p η)q(z)]. (46) Differentiating both sides of (46)withrespecttoz,weobtain z[ ] =(p η)zq (z) + =(p η)zq (z) z[ ] [η + (p η) q (z)] +[λ+(p λ)g(z)][η+(p η)q(z)]. By substituting (47)into(45), we have z( η=(p η)q(z) + (47) (p η) zq (z) (p λ) G (z) +λ+α+β γ+1. (48) The remainder of our proof of Theorem 10 is much akin to that of Theorem 8. We, therefore, choose to omit the details involved. Theorem 11. Let 0 λ, η<p.then C α,β,γ,p C α+1,β,γ,p. (49) Proof. Just as we derived Theorem 9 as a consequence of Theorem 8 by using the equivalence (4), we can also prove Theorem 11 by using Theorem 10 in conjunction with the equivalence (7). Our main results in Theorems 8 11, canthusbeapplied with a view to deducing the following corollaries. Taking γ = 1 in Theorems 8 11 above, we obtain the following corollary. Corollary 12. Let 0 λ, η<p.then S α, (λ) S α+1, (λ), K α, (λ) K α+1, (λ), C α, C α+1,, C α, C α+1,. (50) Remark 13. Taking p = 1 in Corollary 12, weobtainthe results obtained by Gao et al. [14,Theorems1-4]. Taking α=γ=1and β=c(c> p)in Theorems 8 11, we obtain the following corollary.

6 6 ISRN Mathematical Analysis Corollary 14. Let 0 λ, η<p.then S 1,c,p (λ) S 2,c,p (λ), K 1,c,p (λ) K 2,c,p (λ), C 1,c,p C 2,c,p, C 1,c,p C 2,c,p. (51) Remark 15. Taking p = 1 in Corollary 14, weobtainthe results obtained by Gao et al. [14,Corollary1 4]. 3. A Set of Integral-Preserving Properties In this section, we present several integral-preserving properties of the analytic function classes introduced here. In order to obtain the integral-preserving properties involving the integral operator J c,p defined by (13). Theorem 16. Let c be any real number and c> p.iff(z) S α,β,γ,p (λ), thenj c,p(z) S α,β,γ,p (λ), wherej c,pf(z) is defined by (13). Proof. From (13), we have z( J c,p (z)) =(c+p) crα,γ J c,p (z). (52) Let f S α,β,γ,p (λ) and set z( J c,pf (z)) λ = (p λ) q (z), (53) J c,pf (z) whereq(z) is given by (22).By usingidentity (52), we obtain f (z) J c,pf (z) λ+(p λ)q(z) +c =. (54) c+p Differentiating (54) logarithmicallywithrespecttoz, we obtain z( f(z)) f (z) = z( J c,pf(z)) + J c,pf (z) (p λ) zq (z) (p λ) q (z) +λ+c = λ + (p λ) q (z) + (p λ) zq (z) (p λ) q (z) +λ+c. (55) We now choose u=q(z)=u 1 +iu 2,andV =zq (z) = V 1 +iv 2, and define the function Θ by (p λ) V Θ (u, V) =(p λ)u+ (p λ)u+λ+c. (56) It is easy to see that the function Θ(u, V) satisfies the conditions of Lemma 7, and the remaining part of the proof of Theorem 16 is similar to that of Theorem 8. Taking γ = 1 in Theorem 16, we obtain the following corollary. Corollary 17. Let c be any real number and c> p.iff(z) S α, (λ),thenj c,p(z) S α, (λ),wherej c,pf(z) is defined by (13). Theorem 18. Let c be any real number and c> p.iff(z) K α,β,γ,p (λ),thenj c,p (z) K α,β,γ,p (λ),wherej c,p f(z) is defined by (13). Proof. By applying Theorem 16 in conjunction with the equivalence (4), it follows that f (z) K α,β,γ,p (λ) which proves Theorem 18. z p f (z) S α,β,γ,p (λ) J c,p ( z p f (z)) S α,β,γ,p (λ) z p (J c,pf (z)) S α,β,γ,p (λ) J c,p f (z) K α,β,γ,p (λ), (57) Taking γ = 1 in Theorem 18, we obtain the following corollary. Corollary 19. Let c be any real number and c> p.iff(z) K α, (λ),thenj c,p (z) K α, (λ),wherej c,p f(z) is defined by (13). Theorem 20. Let c be any real number and c> p.iff(z) C α,β,γ,p, thenj c,p (z) C α,β,γ,p, wherej c,p f(z) is defined by (13). Proof. Let f C α,β,γ,p. Then there exists a function Ψ S p (λ) such that Re We put z( Ψ (z) so that we have g S α,β,γ,p We next put (λ), Re z( J c,pf (z)) J c,p >η (0 λ,η<p,z U). (58) =Ψ(z), (59) z( >η (z U). (60) =η+(p η)q(z), (61)

7 ISRN Mathematical Analysis 7 where q(z) is given by (22). Thus, by using identity (52), we obtain z( J c,pf (z)) J c,p = (z(j c,pf (z)) ) J c,p =(z[ (z(j c,pf (z)) )] +c (z(j c,pf (z)) )) (z[ J c,p] Differentiating both sides of (66)withrespecttoz,weobtain z[z( J c,pf (z)) ] J c,p =(p η)zq (z) + z[ J c,p] J c,p =(p η)zq (z) [η+(p η)q(z)] (67) +c 1 J c,p ) z[ (z(j c,pf (z)) )] =( J c,p (z(j c,pf (z)) ) +c J ) c,p z[ J c,p] 1 ( J +c). c,p (62) +[λ+(p λ)g(z)][η+(p η)q(z)]. By substituting (67)into(65), we have z( (p η) zq (z) η=(p η)q(z) + (p λ) G (z) +λ+c. (68) The remainder of our proof of Theorem 20 is much akin to that of Theorem 8. We, therefore, choose to omit the details involved. Taking γ = 1 in Theorem 20, we obtain the following corollary. Since g S α,β,γ,p (λ), then from Theorem 16 we have J c,p g(z) S α,β,γ,p (λ),sothatwecanput where z[ J c,p] J c,p = λ + (p λ) G (z), (63) G (z) =g 1 (x, y) + ig 2 (x, y), Re (G (z)) =g 1 (x, y) > 0 Then z( J c,pf (z)) J c,p We thus find from (61)that z[ (z(j c,pf (z)) )] =( J c,p +c(η+(p η)q(z))) ((p λ)g(z) +λ+c) 1. (z U). (64) (65) z( J c,pf (z)) = J c,p [η + (p η) q (z)]. (66) Corollary 21. Let c be any real number and c > p.if f(z) C α,,thenj c,p (z) C α,,wherej c,p f(z) is defined by (13). Theorem 22. Let c be any real number and c> p.iff(z) C α,β,γ,p, thenj c,p(z) C α,β,γ,p, wherej c,pf(z) is defined by (13). Proof. Just as we derived Theorem 18 as a consequence of Theorem 16 by using the equivalence (4), we can also prove Theorem 22 by using Theorem 20 in conjunction with the equivalence (7). Taking γ = 1 in Theorem 22, we obtain the following corollary. Corollary 23. Let c be any real number and c> p.iff(z) C α,β,γ,p, thenj c,p(z) C α,β,γ,p, wherej c,pf(z) is defined by (13). References [1] D. A. Patil and N. K. Thakare, On convex hulls and extreme points of p-valent starlike and convex classes with applications, Bulletin Mathématique de la Société des Sciences Mathématiques de la République Socialiste de Roumanie,vol.27,no.75,pp , [2] S.Owa, Oncertain classes ofp-valent functions with negative coefficients, Simon Stevin. A Quarterly Pure and Applied Mathematics,vol.59,no.4,pp ,1985.

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