Kyungpook Mathematical Journal 2018; 58(2): 243-255  
Certain Subclasses of k–uniformly Functions Involving the Generalized Fractional Differintegral Operator
Tamer Mohamed Seoudy
Department of Mathematics, Faculty of Science, Fayoum University, Fayoum 63514, Egypt, e-mail :
Received: April 25, 2013; Accepted: April 12, 2016; Published online: June 23, 2018.
© Kyungpook Mathematical Journal. All rights reserved.

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We introduce several k–uniformly subclasses of p–valent functions defined by the generalized fractional differintegral operator and investigate various inclusion relationships for these subclasses. Some interesting applications involving certain classes of integral operators are also considered.

Keywords: analytic functions, k-uniformly starlike functions, k-uniformly convex functions, k-uniformly close-to-convex functions, k-uniformly quasi-convex functions, integral operator, Hadamard product, subordination
1. Introduction

Let denote the class of functions of the form:


which are analytic and p–valent in the open unit disk = {z ∈ ℂ : |z| < 1}. If f and g are analytic in , we say that f is subordinate to g, written fg or f(z) ≺ g(z), if there exists a Schwarz function ω, analytic in with ω (0) = 0 and (z)| < 1 (z), such that f(z) = g(ω(z)) (z). In particular, if the function g is univalent in , the above subordination is equivalent to (see [8] and [9]).

For 0 ≤ γ, η < p, k ≥ 0 and z, we define USp*(k;γ), UCp (k; γ), UKp (k; γ, η) and UKp*(k;γ,η) the k–uniformly subclasses of consisting of all analytic functions which are, respectively, p–valent starlike of order γ, p–valent convex of order γ, p–valent close-to-convex of order γ, and type η and p–valent quasi-convex of order γ, and type η as follows:


These subclasses were introduced and studied by Al-Kharsani [1]. We note that

  • (i) US1*(k;γ)=US*(k;γ) and UC1 (k; γ) = UC (k; γ) (0 ≤ γ <1) (see [6] and [20]);

  • (ii) USp*(0;γ)=Sp*(γ)   (0γ<p) (see [12] and [15]);

  • (iii) UCp (0; γ) = Cp (γ) (0 ≤ γ < p) (see [12]);

  • (iv) UKp (0; γ, η) = Kp (γ, η) (0 ≤ γ, η < p) (see [2]);

  • (v) UKp*(0;γ,η)=Kp*(γ,η)(0γ,η<p) (see [10]).

Corresponding to a conic domain Ωp,k,γ defined by


we define the function qp,k,γ (z) which maps onto the conic domain Ωp,k,γ such that 1 ∈ Ωp,k,γ as the following (see [1]):


where u(z)=z-x1-xz, x ∈ (0, 1) and ζ (k) is such that k=coshπζ(z)4ζ(z). By virtue of the properties of the conic domain Ωp,k,γ, we have


Making use of the principal of subordination between analytic functions and the definition of qp,k,γ (z), we may rewrite the subclasses USp*(k;γ), UCp (k; γ), UKp (k; γ, β) and UKp*(k;γ,β) as the following:


Srivastava et al. [23] introduced the following generalized fractional integral and generalized fractional derivative operators as follows(see also [16] and [19]):

Definition 1.1

([23]) For real numbers λ > 0, μ and η, the Saigo hypergeometric fractional integral operator I0,zλ,μ,η:ApAp is defined by

I0,zλ,μ,ηf(z)=z-λ-μΓ(λ)0z(z-t)λ-1F21(λ+μ,-η;λ;1-tz)f(t)   dt,

where the function f(z) is analytic in a simply-connected region of the complex z–plane containing the origin, with the order


and the multiplying of (zt)λ−1 is removed by requiring log (zt) to be real when (zt) > 0.

Definition 1.2

([23]) Under the hypotheses of Definition 1.1, Saigo hypergeometric fractional derivative operator J0,zλ,μ,η:ApAp is defined by

J0,zλ,μ,ηf(z)={1Γ(1-λ)ddz{zλ-μ0z(z-t)-λF21(μ-λ,1-η;1-λ;1-tz)f(t)   dt}(0λ<1),dndznJ0,zλ-μ,μ,ηf(z)(nλ<n+1;n),

where the multiplying of (zt)−λ is removed as in Definition 1.1.

We note that

I0,zλ-λ,ηf(z)=Dz-λf(z)         (λ>0)         and         J0,zλ,λ,ηf(z)=Dzλf(z)         (0λ<1),

where Dz-λ denotes fractional integral operator and Dzλ denotes fractional derivative operator studied by Owa [11].

Recently, Goyal and Prajapat [7] (see also [17] and [18]) introduced the generalized fractional differintegral operator S0,zλ,μ,η:ApAp(p,η,μ<p+1) by

S0,zλ,μ,ηf(z)={Γ(1+p-μ)   Γ(1+p+η-λ)Γ(1+p)Γ(1+p+η-μ)zμJ0,zλ,μ,η(0λ<η+p+1),Γ(1+p-μ)   Γ(1+p+η-λ)Γ(1+p)Γ(1+p+η-μ)zμI0,z-λ,μ,η(-<λ<0).

It is easily seen from a function f of the form (1.1), we have

S0,zλ,μ,ηf(z)=zpF32(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)nan+pzn+p         (zU;p;μ,η;μ<p+1;-<λ<η+p+1),

where qFs (qs + 1; q, s ∈ ℕ0 = ℕ∪{0}) is well known generalized hypergeometric function (see, for details, [13, 22]) and (v)n is the Pochhammer symbol defined, in terms of Gamma function, by


We note that



S0,zλ,λ,0f(z)=Ωz(λ,p)f(z)=Γ(p+1-λ)Γ(p+1)zλDzλf(z)         (-λ<p+1;p;zU),

where the extended fractional differintegral operator Ωz(λ,p) was introduced and studied by Patel and Mishra [14]. The fractional differential operator Ωz(λ,p) with 0 ≤ λ < 1 was investigated by Srivastava and Aouf [21]. The operator Ωz(λ,1)=Ωzλ was introduced by Owa and Srivastava [13];

Upon setting

Gp,η,μλ(z)=zp+n=1(1+p)n(1+p+η-μ)n(1+p-μ)n(1+p+η-λ)nzn+p         (zU;p;μ,η;μ<p+1;-<λ<η+p+1),

we define a new function [Gp,μ,ηλ(z)]-1 by means of the Hadamard product (or convolution):


Tang et al. [24] introduced the linear operator Hp,η,μλ,δ:ApAp as follows:


For f given by (1.1), then from (1.19), we have


by using (1.20), we get




Next, using the operator Hp,η,μλ,δ, we introduce the following k–uniformly subclasses of p–valent functions for η ∈ ℝ, μ < p+1,−∞ < λ < η+ p+1, δ >p, p ∈ ℕ, k ≥ 0 and 0 ≤ γ, ρ < p:


We also note that




In this paper, we investigate several inclusion properties of the classes USp,η,μλ,δ(k;γ),UCp,η,μλ,δ(k;γ),UKp,η,μλ,δ(k;γ,ρ) and UQp,η,μλ,δ(k;γ,ρ) associated with the operator Hp,η,μλ,δ. Some applications involving integral operators are also considered.

2. Inclusion Properties Involving the Operator Hp,η,μλ,δ

In order to prove the main results, we shall need The following lemmas.

Lemma 2.1

([5]) Let h (z) be convex univalent inwith ℜ{αh (z) + β} > 0 (α, β ∈ ℂ). If p (z) is analytic inwith p (0) = h (0), then




Lemma 2.2

([8]) Let h (z) be convex univalent inand let w be analytic inwith ℜ{w (z)} ≥ 0. If p (z) is analytic inand p (0) = h (0), then




Theorem 2.3

Let δ (k + 1)+kp +γ > 0 and (ηλ) (k + 1)+kp +γ > 0. Then,


We first prove that USp,η,μλ,δ+1(k;γ)USp,η,μλ,δ(k;γ). Let fUSp,η,μλ,δ+1(k;γ) and set

p(z)=z(Hp,η,μλ,δf(z))Hp,η,μλ,δf(z)   (zU),

where the function p (z) is analytic in with p (0) = p. Using (1.22), (2.5) and (2.6), we have


Since δ (k + 1) + kp +γ > 0, we see that

{qp,k,γ(z)+δ}>0         (zU).

Applying Lemma 2.1 to (2.7), it follows that p (z) ≺ qp,k,γ (z), that is, fUSp,η,μλ,δ(k;γ). To prove the right part, let fUSp,η,μλ,δ(k;γ) and consider

h(z)=z(Hp,η,μλ+1,δf(z))Hp,η,μλ+1,δf(z)   (zU),

where the function h (z) is analytic in with h (0) = p. Then, by using the arguments similar to those detailed above, together with (1.21), it follows that p (z) ≺ qp,k,γ (z), which implies that fUSp,η,μλ+1,δ(k;γ). Therefore, we complete the proof of Theorem 2.3.

Theorem 2.4

Let δ (k + 1)+kp +γ > 0 and (ηλ) (k + 1)+kp +γ > 0. Then,


Applying (1.27) and Theorem 2.3, we observe that

fUCp,η,μλ,δ+1(k;γ)zfpUSp,η,μλ,δ+1(k;γ)zfpUSp,η,μλ,δ(k;γ)         (by Theorem 2.3),fUCp,η,μλ,δ(k;γ)


fUCp,η,μλ,δ(k;γ)zfpUSp,η,μλ,δ(k;γ)zfpUSp,η,μλ+1,δ(k;γ)         (by Theorem 2.3),fUCp,η,μλ+1,δ(k;γ),

which evidently proves Theorem 2.4.

Next, by using Lemma 2.2, we obtain the following inclusion relation for the class UKp,η,μλ,δ(k;γ,ρ).

Theorem 2.5

Let δ (k + 1)+kp +ρ > 0 and (ηλ) (k + 1)+kp +ρ > 0. Then,


We begin by proving that UKp,η,μλ,δ+1(k;γ,ρ)UKp,η,μλ,δ(k;γ,ρ). Let fUKp,η,μλ,δ+1(k;γ,ρ). Then, from the definition of UKp,η,μλ,δ+1(k;γ,ρ), there exists a function r (z) ∈ USp (k; γ) such that


Choose the function g such that Hp,η,μλ,δ+1g(z)=r(z). Then, gUSp,η,μλ,δ+1(k;γ) and


Now let

p(z)=z(Hp,η,μλ,δf(z))Hp,η,μλ,δ(z)         (zU),

where p (z) is analytic in with p (0) = p. Since gUSp,η,μλ,δ+1(k;γ), by Theorem 2.3, we know that gUSp,η,μλ,δ(k;γ). Let

t(z)=z(Hp,η,μλ,δg(z))Hp,η,μλ,δg(z)         (zU),

where t (z) is analytic in with {t(z)}>kp+ρk+1. Also, from (2.13), we note that

Hp,η,μλ,δzf(z)=Hp,η,μλ,δg(z)   p(z).

Differentiating both sides of (2.15) with respect to z, we obtain


Now using the identity (1.22) and (2.14), we obtain


Since δ (k + 1) + kp +ρ > 0 and {t(z)}>kp+ρk+1, we see that

{t(z)+δ}>0         (zU).

Hence, applying Lemma 2.2, we can show that p (z) ≺ qp,k,γ (z) so that fUKp,η,μλ,δ(k;γ,ρ). For the second part, by using the arguments similar to those detailed above with (1:15), we obtain


Therefore, we complete the proof of Theorem 2.5.

Theorem 2.6

Let δ (k + 1) + kp +ρ > 0and (ηλ) (k + 1) + kp +ρ > 0Then,


Just as we derived Theorem 2.4 as consequence of Theorem 2.3 by using the equivalence (1.27), we can also prove Theorem 2.6 by using Theorem 2.5 and the equivalence (1.28).

3. Inclusion Properties Involving the Integral Operator Fc,p

In this section, we present several integral-preserving properties of the p-valent function classes introduced here. We consider the generalized Libera integral operator Fc,p (f) (see [4] and [3]) defined by

Fc,p(f)(z)=c+pzctc-1f(z)dt   (c>-p).

Theorem 3.1

Let c (k + 1) + kp + γ ≥ 0. If fUSp,η,μλ,δ(k;γ), then Fc,p(f)USp,η,μλ,δ(k;γ).


Let fUSp,η,μλ,δ(k;γ) and set

p(z)=z(Hp,η,μλ,δFc,p(f)(z))Hp,η,μλ,δFc,p(f)(z)         (zU),

where p (z) is analytic in with p (0) = p.

From (3.1), we have


Then, by using (3.2) and (3.3), we obtain


Taking the logarithmic differentiation on both sides of (3.4) and multiplying by z, we have

z(Hp,η,μλ,δf(z))Hp,η,μλ,δf(z)=p(z)+zp(z)p(z)+cqk,γ(z)         (zU).

Hence, by virtue of Lemma 2.1, we conclude that p (z) ≺ qk,γ (z) in , which implies that Fc,p(f)USp,η,μλ,δ(k;γ).

Next, we derive an inclusion property involving Fc,p (f), which is given by the following.

Theorem 3.2

Let c (k + 1) + kp + γ ≥ 0. If fUCp,η,μλ,δ(k;γ), then Fc,p(f)UCp,η,μλ,δ(k;γ).


By applying Theorem 2.5, it follows that

fUCp,η,μλ,δ(k;γ)zfpUSp,η,μλ,δ(k;γ)Fc,p(zfp)USp,η,μλ,δ(k;γ)         (by Theorem 3.1)z(Fc,p(f))pUCp,η,μλ,δ(k;γ)Fc,p(f)UCp,η,μλ,δ(k;γ),

which proves Theorem 3.2.

Theorem 3.3

Let c (k + 1)+kp + ρ ≥ 0. If fUKp,η,μλ,δ(k;γ,ρ), then Fc,p(f)UKp,η,μλ,δ(k;γ,ρ).


Let fUKp,η,μλ,δ(k;γ,ρ). Then, in view of the definition of the class UKp,η,μλ,δ(k;γ,ρ), there exists a function gUSp,η,μλ,δ(k;γ) such that


Thus, we set


where p (z) is analytic in with p (0) = p. Since gUSp,η,μλ,δ(k;γ), we see from Theorem 3.1 that Fc,p(g)USp,η,μλ,δ(k;γ). Let

t(z)=z(Hp,η,μλ,δFc,p(g)(z))Hp,η,μλ,δFc,p(g)(z)         (zU),

where t (z) is analytic in with {t(z)}>kp+ηk+1 Also, from (3.7), we note that


Differentiating both sides of (3.9) with respect to z, we obtain


Now using the identity (3.3) and (3.10), we obtain


Since c (k + 1) + kp + ρ ≥ 0 and {t(z)}>kp+ηk+1, we see that

{t(z)+c}>0         (zU).

Hence, applying Lemma 2.2 to (3.11), we can show that p (z) ≺ qp,k,γ (z) so that Fc,p(f)UKp,η,μλ,δ(k;γ,ρ).

Theorem 3.4

Let c (k + 1)+kp + η ≥ 0. If fUQp,η,μλ,δ(k;γ,ρ), then Fc,p(f)UQp,η,μλ,δ(k;γ,ρ).


Just as we derived Theorem 3.2 as consequence of Theorem 3.1, we easily deduce the integral-preserving property asserted by Theorem 3.4 by using Theorem 3.3.


The author is grateful to the referees for their valuable suggestions.

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