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 : tms00@fayoum.edu.eg
Received: April 25, 2013; Accepted: April 12, 2016; Published online: June 23, 2018.
© Kyungpook Mathematical Journal. All rights reserved.

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Abstract

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:

f(z)=zp+n=1an+pzn+p(p={1,2,3,}),

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:

USp*(k;γ)={fAp:(zf(z)f(z)-γ)>k|zf(z)f(z)-p|},UCp(k;γ)={fAp:(zf(z)f(z)-γ)>k|zf(z)f(z)-p|},UKp(k;γ,η)={fAp:gUSp*(k;η),(xf(z)g(z)-γ)>k|zf(z)g(z)-p|},UKp*(k;γ,η)={fAp:gUCp(k;η),((zf(z))g(z)-γ)>k|(zf(z))g(z)-p|}.

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

Ωp,k,γ={u+iv:u>k(u-p)2+v2+γ},

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]):

qp,k,γ(z)={p+(p-2γ)z1-z(k=0),p-γ1-k2cos{2π(cos-1k)ilog1+z1-z}-k2p-γ1-k2(0<k<1),p+2(p-γ)π2(log1+z1-z)2(k=1),p-γk2-1sin{π2ζ(k)0u(z)kdt1-t21-k2t2}+k2p-γk2-1(k>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

{qp,k,γ(z)}>kp+γk+1.

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:

USp*(k;γ)={fAp:zf(z)f(z)qp,k,γ(z)},UCp(k;γ)={fAp:1+zf(z)f(z)qp,k,γ(z)},UKp(k;γ,η)={fAp:gUSp*(k;η),zf(z)g(z)qp,k,γ(z)},UKp*(k;γ,η)={fAp:gUCp(k;η),(zf(z))g(z)qp,k,γ(z)}.

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

f(z)=O(zɛ)(z0;ɛ>max{0,μ-λ}-1),

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

(v)n=Γ(v+n)Γ(v)={1(n=0)v(v+1)(v+n-1)(n).

We note that

S0,z0,0,0f(z)=f(z),S0,z1,0,0f(z)=zf(z)p

and

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):

Gp,η,μλ(z)*[Gp,η,μλ(z)]-1=zp(1-z)δ+p(δ>-p;zU).

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

Hp,η,μλ,δf(z)=[Gp,η,μλ(z)]-1*f(z).

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

Hp,η,μλ,δf(z)=zp+n=1(δ+p)n(1+p-μ)n(1+p+η-λ)nn!(1+p)n(1+p+η-μ)nan+pzn+p

by using (1.20), we get

z(Hp,η,μλ+1,δf(z))=(p+η-λ)Hp,η,μλ,δf(z)-(η-λ)Hp,η,μλ+1,δf(z)

and

z(Hp,η,μλ,δf(z))=(δ+p)Hp,η,μλ,δ+1f(z)-δHp,η,μλ,δf(z).

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:

USp,η,μλ,δ(k;γ)={fAp:Hp,η,μλ,δf(z)USp*(k;γ);zU},UCp,η,μλ,δ(k;γ)={fAp:Hp,η,μλ,δf(z)UCp(k;γ);zU},UKp,η,μλ,δ(k;γ,ρ)={fAp:Hp,η,μλ,δf(z)UKp(k;γ,ρ);zU},UQp,η,μλ,δ(k;γ,ρ)={fAp:Hp,η,μλ,δf(z)UKp*(k;γ,ρ);zU}.

We also note that

fUSp,η,μλ,δ(k;γ)zfpUCp,η,μλ,δ(k;γ),

and

fUKp,η,μλ,δ(k;γ,ρ)zfpUQp,η,μλ,δ(k;γ,ρ).

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

p(z)+zp(z)αp(z)+βh(z)

implies

p(z)h(z).

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

p(z)+w(z)zp(z)h(z)

implies

p(z)h(z).

Theorem 2.3

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

USp,η,μλ,δ+1(k;γ)USp,η,μλ,δ(k;γ)USp,η,μλ+1,δ(k;γ).
Proof

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

z(Hp,η,μλ,δ+1f(z))Hp,η,μλ,δ+1f(z)=p(z)+zp(z)p(z)+δqp,k,γ(z).

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,

UCp,η,μλ,δ+1(k;γ)UCp,η,μλ,δ(k;γ)UCp,η,μλ+1,δ(k;γ).
Proof

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)

and

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,

UKp,η,μλ,δ+1(k;γ,ρ)UKp,η,μλ,δ(k;γ,ρ)UKp,η,μλ+1,δ(k;γ,ρ).
Proof

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

z(Hp,η,μλ,δ+1f(z))r(z)qp,k,γ(z).

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

z(Hp,η,μλ,δ+1f(z))Hp,η,μλ,δ+1g(z)qp,k,γ(z).

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

z(Hp,η,μλ,δzf(z))Hp,η,μλ,δg(z)=z(Hp,η,μλ,δg(z))Hp,η,μλ,δg(z)p(z)+zp(z)=t(z)p(z)+zp(z).

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

z(Hp,η,μλ,δ+1f(z))Hp,η,μλ,δ+1g(z)=Hp,η,μλ,δ+1zf(z)Hp,η,μλ,δ+1g(z)=z(Hp,η,μλ,δzf(z))+δHp,η,μλ,δzf(z)z(Hp,η,μλ,δg(z))+δHp,η,μλ,δg(z)=z(Hp,η,μλ,δzf(z))Hp,η,μλ,δg(z)+δz(Hp,η,μλ,δf(z))Hp,η,μλ,δg(z)z(Hp,η,μλ,δg(z))Hp,η,μλ,δg(z)+δ=t(z)p(z)+zp(z)+δp(z)t(z)+δ=p(z)+zp(z)t(z)+δ.

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

UKp,η,μλ,δ(k;γ,ρ)UKp,η,μλ+1,δ(k;γ,ρ).

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,

UQp,η,μλ,δ+1(k;γ,ρ)UQp,η,μλ,δ(k;γ,ρ)UQp,η,μλ+1,δ(k;γ,ρ).
Proof

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).

Proof

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

z(Hp,η,μλ,δFc,p(f)(z))=(c+p)Hp,η,μλ,δf(z)-cHp,η,μλ,δFc,p(f)(z).

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

(c+p)Hp,η,μλ,δf(z)Hp,η,μλ,δFc(f)(z)=p(z)+c.

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).

Proof

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).

Proof

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

z(Hp,η,μλ,δf(z))Hp,η,μλ,δg(z)qk,γ(z).

Thus, we set

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

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

Hp,η,μλ,δzFc,p(f)(z)=Hp,η,μλ,δFc,p(g)(z).p(z).

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

z(Hp,η,μλ,δzFc,p(f)(z))Hp,η,μλ,δFc,pg(z)=z(Hp,η,μλ,δFc,p(g)(z))Hp,η,μλ,δFc,p(g)(z)p(z)+zp(z)=t(z)p(z)+zp(z).

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

z(Hp,η,μλ,δf(z))Hp,η,μλ,δg(z)=z(Hp,η,μλ,δzFc,p(f)(z))+cHp,η,μλ,δzFc,p(f)(z)z(Hp,η,μλ,δFc,p(g)(z))+cHp,η,μλ,δFc,p(g)(z)=z(Hp,η,μλ,δzFc,p(f)(z))Hp,η,μλ,δFc,p(g)(z)+cz(Hp,η,μλ,δFc,p(f)(z))Hp,η,μλ,δFc,p(g)(z)z(Hp,η,μλ,δFc,p(g)(z))Hp,η,μλ,δFc,p(g)(z)+c=t(z)p(z)+zp(z)+cp(z)t(z)+c=p(z)+zp(z)t(z)+c.

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).

Proof

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.

Acknowledgements

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

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