Mohamed K. Aouf

Tamer M. Seoudy

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 351 (2013) 787–792Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comComplex analysisSubclasses of p-valent functions of bounded boundaryrotation involving the generalized fractional differintegraloperatorSous-classes de fonctions p-valentes de rotation frontière bornée, relativesà l’opérateur différo-intégral fractionnaire généraliséTamer M. Seoudy a, Mohamed K. Aouf ba Department of Mathematics, Faculty of Science, Fayoum University, Fayoum 63514, Egyptb Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypta r t i c l e i n f o a b s t r a c tArticle history:Received 4 September 2013Accepted after revision 16 October 2013Available online 5 November 2013Presented by the Editorial BoardWe introduce certain subclasses of p-valent functions of bounded boundary rotationinvolving the generalized fractional differintegral operator and investigate various inclusionrelationships for these subclasses. Some interesting applications involving certain classes ofintegral operators are also considered.© 2013 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éNous introduisons certaines sous-classes de fonctions p-valentes de rotation frontièrebornée, relatives à l’opérateur différo-intégral fractionnaire généralisé, et obtenons diversesrelations d’inclusion de ces sous-classes. Quelques applications intéressantes impliquantcertaines classes d’opérateurs intégraux sont également considérées.© 2013 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionLet Ap denote the class of functions of the form:f (z) = zp +∞∑n=1an+p zn+p(p ∈N ={1,2,3, . . .}), (1)which are analytic in the open unit disk U = {z ∈ C: |z| < 1}. If f and g are analytic in U, we say that f is subordinateto g , written f ≺ g or f (z) ≺ g(z), if there exists a Schwarz function ω, analytic in U with ω(0) = 0 and |ω(z)| < 1 (z ∈U),such that f (z) = g(ω(z)) (z ∈ U) (see [5]).Let Vp,k(γ ) be the class of functions g analytic in U satisfying the properties g(0) = p andE-mail addresses: tms00@fayoum.edu.eg (T.M. Seoudy), mkaouf127@yahoo.com (M.K. Aouf).1631-073X/$ – see front matter © 2013 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.http://dx.doi.org/10.1016/j.crma.2013.10.013788 T.M. Seoudy, M.K. Aouf / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 787–7922π∫0∣∣∣∣{g(z)} − γp − γ∣∣∣∣ dθ  kπ (z = reiθ , k  2; 0  γ < p). (2)The class Vp,k(γ ) was introduced by Aouf [1]. We note that:(i) the class V1,k(γ ) = Vk(γ ) was introduced by Padmanabhan and Parvatham [10];(ii) the class V1,k(0) = Vk was introduced by Pinchuk [12];(iii) Vp,2(γ ) = Vp(γ ) is the class of functions with positive real part greater than γ (0  γ < p).From (2), we have g ∈ Vp,k(γ ) if and only if there exist g1, g2 ∈ Vp(γ ) such that:g(z) =(k4+ 12)g1(z) −(k4− 12)g2(z) (z ∈U). (3)It is known [6] that the class Vk(γ ) is a convex set.Making use of Vp,k(γ ), we introduce the subclasses S p,k(γ ) and C p,k(γ ), 0  δ < p, of Ap as follows:S∗p,k(γ ) ={f ∈ Ap: zf′(z)f (z)∈ Vp,k(γ ); z ∈U}, (4)andC p,k(γ ) ={f ∈ Ap: (zf′(z))′f ′(z)∈ Vp,k(γ ); z ∈U}. (5)We note that S p,2(γ ) = S p(γ ) and C p,2(γ ) = C p(γ ) (0  γ < p), where S p(γ ) and C p(γ ) are, respectively, the classes ofp-starlike functions of order γ and p-convex functions of order γ in U (see [8] and [11]).Srivastava et al. [14] introduced the following generalized fractional integral and generalized fractional derivative opera-tors as follows:Definition 1.1. (See [14].) For real numbers λ > 0, μ and η, the Saigo hypergeometric fractional integral operator: Iλ,μ,η0,z :Ap →Ap is defined by:Iλ,μ,η0,z f (z) =z−λ−μΓ (λ)z∫0(z − t)λ−1 2 F1(λ + μ,−η;λ;1 − tz)f (t)dt, (6)where the function f (z) is analytic in a simply-connected region of the complex z-plane containing the origin, with theorder f (z) = O (|z|ε) (z → 0; ε > max{0,μ− λ} − 1), and the multiplying of (z − t)λ−1 is removed by requiring log(z − t) tobe real when (z − t) > 0.Definition 1.2. (See [14].) Under the hypotheses of Definition 1.1, the Saigo hypergeometric fractional derivative operatorJλ,μ,η0,z :Ap →Ap is defined by:Jλ,μ,η0,z f (z) ={ 1Γ (1−λ)ddz{zλ−μ∫ z0 (z − t)−λ 2 F1(μ − λ,1 − η;1 − λ;1 − tz ) f (t)dt}(0  λ < 1),dndzn Jλ−μ,μ,η0,z f (z) (n  λ < n + 1; n ∈N),(7)where the multiplying of (z − t)−λ is removed as in Definition 1.1.We note that Iλ,−λ,η0,z f (z) = D−λz f (z) with λ > 0 and Jλ,λ,η0,z f (z) = Dλz f (z) with 0  λ < 1, where D−λz denotes thefractional integral operator and Dλz denotes the fractional derivative operator studied by Owa [7].Recently, Goyal and Prajapat [4] (see also [13]) introduced the generalized fractional differintegral operator Sλ,μ,η0,z :Ap →Ap (p ∈N; η ∈ R, μ < p + 1; z ∈U) by:Sλ,μ,η0,z f (z) =⎧⎨⎩Γ (1+p−μ)Γ (1+p+η−λ)Γ (1+p)Γ (1+p+η−μ) zμ Jλ,μ,η0,z (0  λ < η + p + 1),Γ (1+p−μ)Γ (1+p+η−λ)Γ (1+p)Γ (1+p+η−μ) zμ I−λ,μ,η0,z (−∞ < λ < 0).(8)It is easily seen from a function f of the form (1), we have:T.M. Seoudy, M.K. Aouf / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 787–792 789Sλ,μ,η0,z f (z) = zp 3 F2(1,1 + p,1 + p + η − μ;1 + p − μ,1 + p + η − λ; z) ∗ f (z)= zp +∞∑n=1(1+p)n(1+p+η−μ)n(1+p−μ)n(1+p+η−λ)n an+p zn+p (μ < p + 1; −∞ < λ < η + p + 1), (9)where q Fs (q  s + 1; q, s ∈ N0 = N∪{0}) is the well-known generalized hypergeometric function (see, for details, [9]) and(v)n is the Pochhammer symbol.Upon settingGλp,η,μ(z) = zp +∞∑n=1(1 + p)n(1 + p + η − μ)n(1 + p − μ)n(1 + p + η − λ)n zn+p (μ < p + 1; −∞ < λ < η + p + 1), (10)we define a new function [Gλp,μ,η(z)]−1 by means of the Hadamard product (or convolution):Gλp,η,μ(z) ∗[Gλp,η,μ(z)]−1 = zp(1 − z)δ+p (δ > −p; z ∈U). (11)Tang et al. [15] introduced the linear operator Hλ,δp,η,μ :Ap →Ap as follows:Hλ,δp,η,μ f (z) =[Gλp,η,μ(z)]−1 ∗ f (z). (12)For f ∈Ap given by (1), then from (12), we have:Hλ,δp,η,μ f (z) = zp +∞∑n=1(δ + p)n(1 + p − μ)n(1 + p + η − λ)n(1 + p)n(1 + p + η − μ)nn! an+p zn+p (13)by using (13), we get:z(Hλ+1,δp,η,μ f (z))′ = (p + η − λ)Hλ,δp,η,μ f (z) − (η − λ)Hλ+1,δp,η,μ f (z) (14)andz(Hλ,δp,η,μ f (z))′ = (δ + p)Hλ,δ+1p,η,μ f (z) − δHλ,δp,η,μ f (z). (15)Next, using the operator Hλ,δp,η,μ , we introduce the subclasses Sλ,δp,k(η,μ;γ ) and Cλ,δp,k(η,μ;γ ) of Ap for μ,η ∈ R, μ <p + 1, −∞ < λ < η + p + 1, δ > −p, p ∈N, k  0 and 0  γ ,ρ < p:Sλ,δp,k(η,μ;γ ) ={f ∈ Ap: Hλ,δp,η,μ f (z) ∈ S∗p,k(γ ) (z ∈U)}, (16)andCλ,δp,k(p, η,μ;γ ) ={f ∈ Ap: Hλ,δp,η,μ f (z) ∈ C p,k(γ ) (z ∈U)}. (17)We also note that:f ∈ Sλ,δp,k(η,μ;γ ) ⇔zf ′p∈ Cλ,δp,k(p, η,μ;γ ). (18)In this paper, we investigate several inclusion properties of the classes Sλ,δp,k(η,μ;γ ) and Cλ,δp,k(p, η,μ;γ ) associated withthe operator Hλ,δp,η,μ . Some applications involving integral operators are also considered.2. Inclusion properties involving the operator Hλ,δp,η,μIn order to prove the main results, we shall need the following lemmas.Lemma 2.1. (See [3].) Let h be convex univalent in U with {αh(z) + β} > 0 (α,β ∈ C). If p is analytic in U with p(0) = h(0), then:p(z) + zp′(z)αp(z) + β ≺ h(z) ⇒ p(z) ≺ h(z). (19)Theorem 2.2. Let k  2, δ  0 and η  λ. Then,Sλ,δ+1p,k (η,μ;γ ) ⊂ Sλ,δp,k(η,μ;γ ) ⊂ Sλ+1,δp,k (η,μ;γ ).790 T.M. Seoudy, M.K. Aouf / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 787–792Proof. First of all, we will show that Sλ,δ+1p,k (η,μ;γ ) ⊂ Sλ,δp,k(η,μ;γ ). Let f ∈ Sλ,δ+1p,k (η,μ;γ ) and set:g(z) = z(Hλ,δp,η,μ f (z))′Hλ,δp,η,μ f (z)(z ∈U), (20)where the function g is analytic in U with g(0) = p. Using (15) and (20), we have:z(Hλ,δ+1p,η,μ f (z))′Hλ,δ+1p,η,μ f (z)= g(z) + zg′(z)g(z) + δ ∈ Vp,k(γ ) (k  2; 0  γ < p; z ∈U). (21)Our aim is to show that g(z) ∈ Vp,k(γ ). If g(z) + zg′(z)g(z)+δ ∈ Vp,k(γ ), then there exist two functions h1,h2 ∈ Vp(γ ) such that:g(z) + zg′(z)g(z) + δ =(k4+ 12)h1(z) −(k4− 12)h2(z).Now let:hi(z) = gi(z) + zg′i(z)gi(z) + δ ≺p + (p − 2γ )z1 − z (i = 1,2; 0  γ < p), (22)since δ  0, we see that:{p + (p − 2γ )z1 − z + δ}> 0 (0  γ < p; z ∈U).Applying Lemma 2.1 to (22), it follows that:gi(z) ≺ p + (p − 2γ )z1 − z (i = 1,2; 0  γ < p).This means that (gi(z)) > γ , i = 1,2. Now, if:g(z) =(k4+ 12)g1(z) −(k4− 12)g2(z),then g ∈ Vp,k(γ ), that is, f ∈ Sλ,δp,k(η,μ;γ ). To prove the second part, let f ∈ Sλ,δp,k(η,μ;γ ) and put:q(z) = z(Hλ+1,δp,η,μ f (z))′Hλ+1,δp,η,μ f (z)(z ∈U),where q is analytic with q(0) = p. Then, by using the arguments similar to those detailed above with (14), it follows thatq ∈ Vp,k(γ ) in U, which implies that f ∈ Sλ+1,δp,k (η,μ;γ ). This proves Theorem 2.2. Theorem 2.3. Let k  2, δ  0 and η  λ. Then,Cλ,δ+1p,k (η,μ;γ ) ⊂ Cλ,δp,k(η,μ;γ ) ⊂ Cλ+1,δp,k (η,μ;γ ).Proof. Applying (18) and Theorem 2.2, we observe that:f ∈ Cλ,δ+1p,k (η,μ;γ ) ⇐⇒zf ′p∈ Sλ,δ+1p,k (η,μ;γ ) ⇒zf ′p∈ Sλ,δp,k(η,μ;γ ) ⇐⇒ f ∈ Cλ,δp,k(η,μ;γ ),andf ∈ Cλ,δp,k(η,μ;γ ) ⇐⇒zf ′p∈ Sλ,δp,k(η,μ;γ ) ⇒zf ′p∈ Sλ+1,δp,k (η,μ;γ ) ⇐⇒ f ∈ Cλ+1,δp,k (η,μ;γ ),which evidently proves Theorem 2.3. T.M. Seoudy, M.K. Aouf / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 787–792 7913. Inclusion properties involving the integral operator F p,cIn this section, we consider the generalized Libera integral operator F p,c( f ) (see [2]) defined by:F p,c( f )(z) = c+pzc∫tc−1 f (z)dt (c > −p). (23)Theorem 3.1. If f ∈ Sλ,δp,k(η,μ;γ ), then F p,c( f ) ∈ Sλ,δp,k(η,μ;γ ) (k  2, c  0).Proof. Let f ∈ Sλ,δp,k(η,μ;γ ) and set:g(z) = z(Hλ,δp,η,μF p,c( f )(z))′Hλ,δp,η,μF p,c( f )(z)(z ∈ U), (24)where g is analytic in U with g(0) = p. From (23), we have:z(Hλ,δp,η,μF p,c( f )(z))′ = (c + p)Hλ,δp,η,μ f (z) − cHλ,δp,η,μF p,c( f )(z). (25)Then, by using (24) and (25), we obtain:(c + p) Hλ,δp,η,μ f (z)Hλ,δp,η,μF p,c( f )(z)= g(z) + c. (26)Taking the logarithmic differentiation on both sides of (26) and multiplying by z, we have:g(z) + zg′(z)g(z) + c =z(Hλ,δp,η,μ f (z))′Hλ,δp,η,μ f (z)∈ Vp,k(γ ) (k  2; 0  γ < p; z ∈U). (27)Now the remaining part of Theorem 3.2 follows by employing the techniques of the first part that we used in provingTheorem 2.2 above. Next, we derive an inclusion property involving F p,c( f ), which is given by the following theorem.Theorem 3.2. If f ∈ Cλ,δp,k(η,μ;γ ), then F p,c( f ) ∈ Cλ,δp,k(η,μ;γ ) (k  2; c  0).Proof. By applying Theorem 3.1, it follows that:f ∈ Cλ,δ+1p,k (η,μ;γ ) ⇐⇒zf ′p∈ Sλ,δp,k(η,μ;γ ) ⇒ F p,c(zf ′p)∈ Sλ,δp,k(η,μ;γ )⇐⇒ z(F p,c( f )(z))′p∈ Sλ,δp,k(η,μ;γ ) ⇐⇒ F p,c( f ) ∈ Cλ,δp,k(η,μ;γ )which proves Theorem 3.2. Theorem 3.3. The function f belongs to Sλ,δp,k(η,μ;γ ) (or Cλ,δp,k(η,μ;γ )) if and only if χ defined byχ(z) = δ+pzδz∫0tδ−1 f (t)dt (δ > −p) (28)belongs to Sλ,δ+1p,k (η,μ;γ ) (or Cλ,δ+1p,k (η,μ;γ )).Proof. From (28), we have:(δ + p) f (z) = δχ(z) + zχ ′(z). (29)Using (15) and (29), we can write:(δ + p)Hλ,δp,η,μ f (z) = δHλ,δp,η,μχ(z) + z(Hλ,δp,η,μχ(z))′ = (δ + p)Hλ,δ+1p,η,μχ(z).Therefore Hλ,δp,η,μ f (z) = Hλ,δ+1p,η,μχ(z) and this proves our result. Similarly we can prove the following theorem.792 T.M. Seoudy, M.K. Aouf / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 787–792Theorem 3.4. The function f belongs to Sλ+1,δp,k (η,μ;γ ) (or Cλ+1,δp,k (η,μ;γ )) if and only if ψ given by:ψ(z) = p+η−λzη−λz∫0tη−λ−1 f (t)dt (η > λ) (30)belongs to Sλ,δp,k(η,μ;γ ) (or Cλ,δp,k(η,μ;γ )).References[1] M.K. Aouf, A generalization of functions with real part bounded in the mean on the unit disc, Math. Jpn. 33 (2) (1988) 175–182.[2] N.E. Cho, O.S. Kwon, H.M. Srivastava, Inclusion relationships and argument properties for certain subclasses of multivalent functions associated with afamily of linear operators, J. Math. Anal. Appl. 292 (2004) 470–483.[3] P. Eenigenburg, S.S. Miller, P.T. Mocanu, M.O. Reade, On a Briot–Bouquet Differential Subordination, General Inequalities, vol. 3, Birkhäuser, Basel,Switzerland, 1983, pp. 339–348.[4] G.P. Goyal, J.K. Prajapat, A new class of analytic p-valent functions with negative coefficients and fractional calculus operators, Tamsui Oxford Univ. J.Math. Sci. 20 (2) (2004) 175–186.[5] S.S. Miller, P.T. Mocanu, Differential Subordination: Theory and Applications, in: Series in Pure and Applied Mathematics, vol. 225, Marcel Dekker, NewYork, 2000.[6] K.I. Noor, On subclasses of close-to-convex functions of higher order, Int. J. Math. Math. Sci. 15 (1992) 279–290.[7] S. Owa, On the distortion theorems I, Kyungpook Math. J. 18 (1978) 53–59.[8] S. Owa, On new classes of p-valent function with negative coefficients, Simon Stevin 59 (4) (1985) 385–402.[9] S. Owa, H.M. Srivastava, Univalent and starlike generalized hypergeometric functions, Can. J. Math. 39 (1987) 1057–1077.[10] K.S. Padmanabhan, R. Parvatham, Properties of a class of functions with bounded boundary rotation, Ann. Pol. Math. 31 (1975) 311–323.[11] D.A. Patil, N.K. Thakare, On convex hulls and extreme points of p-valent starlike and convex classes with applications, Bull. Math. Soc. Sci. Math. Roum.27 (1983) 145–160.[12] B. Pinchuk, Functions with bounded boundary rotation, Isr. J. Math. 10 (1971) 7–16.[13] J.K. Prajapat, M.K. Aouf, Majorization problem for certain class of p-valently analytic function defined by generalized fractional differintegral operator,Comput. Math. Appl. 63 (1) (2012) 42–47.[14] H.M. Srivastava, M. Saigo, S. Owa, A class of distortion theorems involving certain operators of fractional calculus, J. Math. Anal. Appl. 131 (1988)412–420.[15] H. Tang, G.-T. Deng, S.-H. Li, M.K. Aouf, Inclusion results for certain subclasses of spiral-like multivalent functions involving a generalized fractionaldifferintegral operator, Integral Transforms Spec. Funct. 24 (11) (2013) 873–883.

Subclasses of <i>p</i>-valent functions of bounded boundary rotation involving the generalized fractional differintegral operator

Seoudy, Tamer M.

Aouf, Mohamed K.

Numérisation de Documents Anciens Mathématiques

https://comptes-rendus.academie-sciences.fr/mathematique/item/10.1016/j.crma.2013.10.013.pdf

Subclasses of <i>p</i>-valent functions of bounded boundary rotation involving the generalized fractional differintegral operator

Abstract

Similar works

Full text

Available Versions

Comptes Rendus Mathématique

Numérisation de Documents Anciens Mathématiques