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eISSN 0454-8124
pISSN 1225-6951

### Article

Kyungpook Mathematical Journal 2016; 56(2): 597-610

Published online June 1, 2016

### Some Inclusion Properties of New Subclass of Starlike and Convex Functions associated with Hohlov Operator

Janusz Sokól1, Gangadharan Murugusundaramoorthy2, Thilagavathi Kothandabani2

Department of Mathematics, Rzesz´ow University of Technology, Al. Powsta´nc´ow Warszawy 12, 35-959 Rzesz´ow, Poland1
School of advanced Sciences, VIT University, Vellore - 632014, India2

Received: March 7, 2014; Accepted: November 3, 2015

### Abstract

For a sufficiently adequate special case of the Dziok-Srivastava linear operator defined by means of the Hadamard product (or convolution) with Srivastava-Wright convolution operator, the authors investigate several mapping properties involving various subclasses of analytic and univalent functions, G(λ, α) and M(λ, α). Furthermore we discuss some inclusion relations for these subclasses to be in the classes of k-uniformly convex and k-starlike functions.

Keywords: Srivastava-Wright convolution operator, ,Starlike functions, Convex functions, Uniformly Starlike functions, ,Uniformly Convex functions, Hadamard product, Hohlov operator, ,Gaussian hypergeometric ,function, Dziok-Srivasta

### 1. Introduction

Let ℋ be the class of functions analytic in the unit disk = {z ∈ ℂ : |z| < 1}. Let be the class of functions f ∈ ℋ of the form

f(z)=z+n=2anzn         zU.

As usual, we denote by the subclass of consisting of functions which are normalized by f(0) = 0 = f′ (0) − 1 and also univalent in . Denote by the subclass of consisting of functions of the form

f(z)=z-n=2anzn,an>0.

A function f is said to be starlike of order α (0 ≤ α < 1), if and only if ℜ(zf′ (z)/f(z)) > α (z). This function class is denoted by * (α). We also write *(0) =: *, where * denotes the class of functions f that f( ) is starlike with respect to the origin. A function f is said to be convex of order α (0 ≤ α < 1) if and only if ℜ(1 + zf″ (z)/f′ (z)) > α (z). This class is denoted by (α). Further, = (0), the well-known standard class of convex functions. It is an established fact that f(α) ⇔ zf′ ∈ *(α).

Furthermore, we denote by k and k, (0 ≤ k < ), two interesting subclasses of consisting respectively of functions which are k-uniformly convex and k-starlike in . Namely, we have for 0 ≤ k <

k-UCV:={fS:(1+zf(z)f(z))>k|zf(z)f(z)|,         (zU)}

and

k-SJ:={fS:(zf(z)f(z))>k|zf(z)f(z)-1|,         (zU)}.

The class 1 − was defined and discussed by Goodman [7]. Further the classes k and k were introduced and its geometric definitions, connections with the conic domains were investigated in [11, 12].

The Gaussian hypergeometric function F(a, b; c, z) given by

F(a,b;c;z)=n=0(a)n(b)n(c)n(1)nzn         (zU)

is the solution of the homogenous hypergeometric differential equation

z(1-z)w(z)+[c-(a+b+1)z]w(z)-abw(z)=0

and has rich applications in various fields such as conformal mappings, quasi conformal theory, continued fractions, and so on. Here, a, b, c are complex numbers such that c ≠ 0,−1, −2, −3, …, (a)0 = 1 for a ≠ 0, and for each positive integer n, (a)n = a(a + 1)(a + 2) … (a + n − 1) is the Pochhammer symbol. In the case of c = −k, k = 0, 1, 2, … , the function F(a, b; c; z) is defined if a = −j or b = −j where jk. We refer to [2, 14] and references therein for some important results.

Also for functions f given by (1.1) and g given by g(z)=z+n=2bnzn, we define the Hadamard product (or convolution) of f and g by

(f*g)(z)=z+n=2anbnzn,         zU.

In terms of the Hadamard product (or convolution), the Dziok-Srivastava linear operator involving the generalized hypergeometric function, was introduced and studied systematically by Dziok and Srivastava [6, 5] and (subsequently) by many other authors. Here, in our present investigation, we recall a familiar convolution operator Ia,b,c due to Hohlov [8], which indeed is a very specialized case of the widely- (and extensively-) investigated Dziok-Srivastava operator. For f, we recall the operator Ia,b,c(f) of which maps into itself defined by means of Hadamard product as

Ia,b,c(f)(z)=zF(a,b;c;z)*f(z).

Therefore, for a function f defined by (1.1), we have

Ia,b,c(f)(z)=z+n=2(a)n-1b(n-1)(c)n-1(1)n-1anzn.

The Hohlov operator Ia,b,c (which has been emphasized upon in this paper) is a very specialized case of the Dziok-Srivastava linear operator [5, 6] which, in turn, is contained in the Srivastava-Wright convolution operator [15] (see also [9]). It is the Srivastava-Wright convolution operator [15] (see also [9]) that is defined by using the Fox-Wright generalized hypergeometric function.

Using the integral representation,

F(a,b;c;z)=Γ(c)Γ(b)Γ(c-b)01tb-1(1-t)c-b-1dt(1-tz)a,(c)>(b)>0,

we can write

Ia,b,c(f)(z)=(Γ(c)Γ(b)Γ(c-b)01tb-1(1-t)c-b-1f(tz)tdt)*z(1-tz)a.

When f(z) equals the convex function z/(1 − z), then the operator Ia,b,c(f) in this case becomes zF(a, b; c; z). If a = 1, b = 1+δ, c = 2+δ with ℜ(δ) > −1, then the convolution operator Ia,b,c(f) turns into Bernardi operator

Bf(z)=Ia,b,c(f)(z)=1+δzδ01tδ-1f(t)dt.

Indeed, I1,1,2(f) and I1,2,3(f) are known as Alexander and Libera operators, respectively.

Let us denote (see [11], [12])

P1(k)={8(arccosk)2π2(1-k2)for0k<1,8/π2fork=1,π24t(1+t)(k2-1)K2(t)fork>1,

where t ∈ (0, 1) is determined by k = cosh(πK′(t)/[4K(t)]), K is the Legendre’s complete Elliptic integral of the first kind

K(t)=01dx(1-x2)(1-t2x2)

and K(t)=K(1-t2) is the complementary integral of K(t). Let Ωk be a domain such that 1 ∈ Ωk and

Ωk={w=u+iv:         u2=k2(u-1)2+k2v2},0k<.

The domain Ωk is elliptic for k > 1, hyperbolic when 0 < k < 1, parabolic when k = 1, and a right half-plane when k = 0. If p is an analytic function with p(0) = 1 which maps the unit disc conformally onto the region Ωk, then P1(k) = p′(0). P1(k) is strictly decreasing function of the variable k and it values are included in the interval (0, 2].

Let f be of the form (1.1). If fk, then the following coefficient inequalities hold true (cf. [11]):

an(P1(k))n-1n!,         n{1}.

Similarly, if f of the form (1.1) belongs to the class k, then (cf., [12])

an(P1(k))n-1(n-1)!,         n{1}.

A function f is said to be in the class ℛτ (A,B), (τ ∈ ℂ{0}, −1 ≤ B < A ≤ 1), if it satisfies the inequality

|f(z)-1(A-B)τ-B[f(z)-1]|<1         (zU).

The class ℛτ (A,B) was introduced earlier by Dixit and Pal [4]. Two of the many interesting subclasses of the class ℛτ (A,B) are worthy of mention here. First of all, by setting

τ=eiηcos η(-π/2<η<π/2),A=1-2β(0β<1)and B=-1,

the class ℛτ (A,B) reduces essentially to the class ℛη(β) introduced and studied by Ponnusamy and Rønning [14], where

η(β)={fA:(eiη(f(z)-β))>0         zU}.

Secondly, if we put

τ=1,A=βand B=-β(0<β1),

we obtain the class of functions f satisfying the inequality

|f(z)-1f(z)+1|<β,         zU

which was studied by (among others) Padmanabhan [13] and Caplinger and Causey [3].

Motivated by the earlier work of Srivastava et al. [16], in this paper we introduce two new subclasses of namely G(λ, α) and M(λ, α) to obtain coefficient bounds and to discuss some inclusion properties involving Hohlov operator.

For some α(0 ≤ α < 1) and λ(0 ≤ λ < 1), we let G(λ, α) and M(λ, α) be two new subclass of consisting of functions of the form (1.1) satisfying the analytic criteria

G(λ,α):={fS:((1-λ)f(z)zf(z)+λ)>α,zU},M(λ,α):={fS:(f(z)+λzf(z)f(z)+zf(z))>α,zU}.

Also denote G*(λ, α) = G(λ, α)∩ and M*(λ, α) = M(λ, α)∩ , the subclasses of defined in (1.2).

### 2. Coefficient Bounds

In his section, we obtain the necessary and sufficient conditions for functions fG(λ, α) and fM(λ, α).

### Lemma 2.1

A function fof the form (1.1) belongs to the class G(λ, α) if f(z)/(zf′(z)) ∈ ℋ and if

n=2(1-nλ-λ-αn)an1-α.
Proof

It is suffices to show that the values for (1-λ)f(z)zf(z)+λ lie in a circle centered at ω = 1 whose radius is 1 − α. We have

|(1-λ)f(z)zf(z)+(λ-1)|=n=2(1-λ+nλ-n)anznz+n=2nanznn=2(1-λ+nλ-n)anzn-11-n=2nanzn-1n=2(1-λ+nλ-n)an1-n=2nan.

The last expression is bounded above by 1 − α if

n=2(1-λ+nλ-n)an(1-α)(1-n=2nan),

which is equivalent to

n=2(1+nλ-λ-αn)an(1-α).

But (2.6) is true by hypothesis. Hence

|(1-λ)f(z)zf(z)+(λ-1)|1-α

and the theorem is proved.

### Corollary 2.2

A function fof the form (1.1) belongs to the class M(λ, α) if f(z)/(zf′(z)) ∈ ℋ and if

n=2n(1+nλ-αn)an1-α.
Proof

It is well known that fM(λ, α) if and only if zf′ ∈ G(λ, α). Since zf=z+n=2nanzn we may replace an with nan in Lemma 2.1. For functions in the converse of Lemma 2.1 is also true.

### Lemma 2.3

A function fbelongs to the class G*(λ, α) if and only if f(z)/(zf′(z)) ∈ ℋ and if

n=2(1+nλ-λ-αn)an1-α.
Proof

In view of Lemma(2.1), it suffices to show the only if part. Assume that

((1-λ)f(z)zf(z)+λ)={z-n=2((1-λ)+nλ)anzn-1z-n=2nanzn-1}>α,(z<1).

Choose values of z on the real axis so that ((1-λ)f(z)zf(z)+λ) is real. Upon clearing the denominator in (2.8) and letting z → 1 through real values, we obtain

1-n=2(1-λ+nλ)anα(1-n=2nan).

Thus n=2(1+nλ-λ-αn)an1-α, and the proof is complete.

### Corollary 2.4

A function fof the form (1.1) belongs to the class M*(λ, α) if and only if f(z)/(zf′(z)) ∈ ℋ and

n=2n(1+nλ-λ-αn)an1-α.

### 3. Inclusion Properties

Making use of the following lemma, we will study the action of the hypergeometric function on the classes k, k.

### Lemma 3.5

[4] If f ∈ ℛτ (A,B) is of form (1.1), then

an(A-B)τn,         n{1}.

The result is sharp.

### Theorem 3.6

Let a, b ∈ ℂ{0}, |a| ≠ 1, |b| ≠ 1. Also, let c be a real number such that c > |a| + |b| + 1. If f ∈ ℛτ (A,B), Ia,b,c(f)/(zIa,b,c(f))and if the inequality

Γ(c)Γ(c-a-b-1)Γ(c-a)Γ(c-b)[(λ-α)(c-a-b-1)+(1-λ)(a-1)(b-1)](1-α)(1(A-B)τ+1)+(1-λ)c-1(a-1)(b-1)

is satisfied, then Ia, b, c(f) ∈ G(λ, α).

Proof

Let f be of the form (1.1) belong to the class ℛτ (A,B). By virtue of Lemma 2.1, it suffices to show that

n=2(1+nλ-λ-nα)|(a)n-1(b)n-1(c)n-1(1)n-1an|1-α.

Taking into account the inequality (3.4) and the relation |(a)n−1| ≤ (|a|)n−1, we deduce that

n=2(1+nλ-λ-nα)|(a)n-1(b)n-1(c)n-1(1)n-1an|(A-B)τ(λ-α)n=2|(a)n-1(b)n-1(c)n-1(1)n-1|+(A-B)τ(1-λ)n=2(a)n-1(b)n-1(c)n-1(1)n(A-B)τ{(λ-α)n=2(a)n-1(b)n-1(c)n-1(1)n-1+(c-1)(1-λ)(a-1)(b-1)n=2(a-1)n(b-1)n(c-1)n(1)n}=(λ-α)(A-B)τ(F(a,b,c;1)-1)+(A-B)τ(1-λ)(c-1)(a-1)(b-1)(F(a-1,b-1,c-1;1)-(a-1)(b-1)c-1-1),

where we use the relation

(a)n=a(a+1)n-1.

The proof now follows by an application of Gauss summation theorem and (3.5).

### Theorem 3.7

Let a, b ∈ ℂ{0}, |a| 1, |b| 1. Also, let c be a real number such that c > |a| + |b| + 2. If f, Ia,b,c(f)/(zIa,b,c(f))and if the inequality

Γ(c)Γ(c-a-b-1)Γ(c-a)Γ(c-b)[1-α+(λ-α)(a)2(b)2(1-α)(c-a-b-2)2+ab(2λ-3α+1)c-a-b-1(c-a-b-1)]2(1-α)

is satisfied, then Ia, b, c(f) ∈ G(λ, α).

Proof

Let f be of the form (1.1) belong to the class . By virtue of Lemma 2.1, it suffices to show that

S(a,b,c,λ,α):=n=2(1+nλ-λ-nα)(a)n-1(b)n-1(c)n-1(1)n-1an1-α.

Applying the well known estimate for the coefficients of the functions f, due to de Branges [1], we need to show that

n=2n(1+nλ-λ-nα)|(a)n-1(b)n-1(c)n-1(1)n-1|1-α.

Taking into account the inequality |(a)n−1| ≤ (|a|)n−1, we deduce that

S(a,b,c,λ,α)n=2(n2(λ-α+n(1-λ))(a)n-1(b)n-1(c)n-1(1)n-1

writing n = (n−1)+1, and n2 = (n−1)(n−2)+3(n−1)+1, we can rewrite the above term as

S(a,b,c,λ,α)(λ-α)n=2(n-1)(n-2)(a)n-1(b)n-1(c)n-1(1)n-1+(2λ-3α+1)n=2(n-1)(a)n-1(b)n-1(c)n-1(1)n-1+(1-α)n=2(a)n-1(b)n-1(c)n-1(1)n-1.

Repeatedly using the relation given in (3.6),

S(a,b,c,λ,α)(λ-α)n=3(a)n-1(b)n-1(c)n-1(1)n-3+(2λ-3α+1)n=2(a)n-1(b)n-1(c)n-1(1)n-2+(1-α)n=2(a)n-1(b)n-1(c)n-1(1)n-1.

The inequality (3.7) now follows by applying Gauss summation theorem and by the hypothesis.

### Theorem 3.8

Let a, b ∈ ℂ{0}. Also, let c be a real number and P1 = P1(k) be given by (1.7). If fk, for some k (0 ≤ k < ), and the inequality

(λ-α)3F2(a,b,P1;c,1;1)+(1-λ)F32(a,b,P1;c,2;1)2(1-α)

is satisfied, then Ia, b, c(f) ∈ G(λ, α).

Proof

Let f be given by (1.1). By (2.3), to show Ia, b, c(f) ∈ M(λ, α), it is sufficient to prove that

n=2(1+nλ-λ-nα)(a)n-1(b)n-1(c)n-1(1)n-1an1-α.

We will repeat the method of proving used in the proof of Theorem 1. Applying the estimates for the coefficients given by (1.8), and making use of the relations (3.6) and |(a)n| ≤ (|a|)n, we get

n=2(1+nλ-λ-nα)(a)n-1(b)n-1(c)n-1(1)n-1ann=2[n(λ-α)+(1-λ)](a)n-1(b)n-1(P1)n-1(c)n-1(1)n-1(1)n=(λ-α)n=2(a)n-1(b)n-1(P1)n-1(c)n-1(1)n-1(1)n-1+(1-λ)n=2(a)n-1(b)n-1(P1)n-1(c)n-1(1)n-1(1)n=(λ-α)[F32(a,b,P1;c,1;1)-1]+(1-λ)[F32(a,b,P1;c,2;1)-1]1-α

provided the condition (3.8) is satisfied.

### Theorem 3.9

Let a, b ∈ ℂ{0}. Also, let c be a real number such that c > |a| + |b| + 1. If f ∈ ℛτ (A,B), and if the inequality

Γ(c)Γ(c-a-b-1)Γ(c-a)Γ(c-b)[(λ-α)ab+(1-α)(c-a-b-1)](1-α)(1(A-B)τ+1)

is satisfied, then Ia, b, c(f) ∈ M(λ, α).

Proof

Let f be of the form (1.1) belong to the class ℛτ (A,B). By virtue of Lemma 3.6, it suffices to show that

n=2n(1+nλ-λ-αn)|(a)n-1(b)n-1(c)n-1(1)n-1an|1-α.

Taking into account the inequality (3.4) and the relation |(a)n−1| ≤ (|a|)n−1, we deduce that

n=2n(1+nλ-λ-αn)|(a)n-1(b)n-1(c)n-1(1)n-1an|(A-B)τ(λ-α)n=2n|(a)n-1(b)n-1(c)n-1(1)n-1|+(A-B)τ(1-α)n=2(a)n-1(b)n-1(c)n-1(1)n(A-B)τ{(λ-α)n=2(a)n-1(b)n-1(c)n-1(1)n-2+(1-α)n=2(a)n-1(b)n-1(c)n-1(1)n-1}=(A-B)τ{(λ-α)abcF(1+a,1+b,1+c;1)(1-α)(F(a,b,c;1)-1)}=(A-B)τ{(λ-α)abcΓ(c-a-b-1)Γ(c+1)Γ(c-a)Γ(c-a)+(1-α){Γ(c-a-b)Γ(c)Γ(c-a)Γ(c-b)-1}}=(A-B)τΓ(c-a-b-1)Γ(c)Γ(c-a)Γ(c-a){(λ-α)ab+(1-α)(c-a-b-1)}-(A-B)τ(1-α)=(A-B)τ{(1-α){1(A-B)τ+1}}-(A-B)τ(1-α)(1-α)

provided the condition (3.10) is satisfied.

### Theorem 3.10

Let a, b ∈ ℂ{0}. Also, let c be a real number and P1 = P1(k) be given by (1.7). If, for some k (0 ≤ k < ), fk, and the inequality

(λ-α)abP1cF32(1+a,1+b,1+P1;1+c,2;1)+(1-α)F32(a,b,P1;c,1;1)2(1-α)

is satisfied, then Ia, b, c(f) ∈ M(λ, α).

Proof

Let f be given by (1.1). By (2.3), to show Ia, b, c(f) ∈ M(λ, α), it is sufficient to prove that

n=2n(1+nλ-λ-αn)|(a)n-1(b)n-1(c)n-1(1)n-1an|1-α.

We will repeat the method of proving used in the proof of the first Theorem. Applying the estimates for the coefficients given by (1.8), and making use of the relations (3.6) and |(a)n| ≤ (|a|)n, we get

n=2n[(1+nλ-λ-αn)]|(a)n-1(b)n-1(c)n-1(1)n-1an|n=2n[n(λ-α)+(1-λ)](a)n-1(b)n-1(P1)n-1(c)n-1(1)n-1(1)n-1=(λ-α)n=2abP1c(1+a)n-2(1+b)n-2(1+P1)n-2(1+c)n-2(1)n-2(2)n-2+(1-α)n=2(a)n-1(b)n-1(P1)n-1(c)n-1(1)n-1(1)n-1=(λ-α)abP1c[F32(1+a,1+b,1+P1;1+c,2;1)]+(1-α)[F32(a,b,P1;c,1;1)-1]1-α

provided the condition (3.12) is satisfied.

### Theorem 3.11

Let a, b ∈ ℂ{0}. Also, let c be a real number and P1 = P1(k) be given by (1.7). If fk, for some k (0 ≤ k < ), and the inequality

(λ-α)abP1cF32(1+a,1+b,1+P1;1+c,1;1)+(1+λ-2α)abP1cF32(1+a,1+b,1+P1;1+c,2;1)+(1-α)F32(a,b,P1;c,1;1)2(1-α).

is satisfied, then Ia, b, c(f) ∈ M(λ, α).

Proof

Let f be given by (1.1). We will repeat the method of proving used in the proof of Theorem 3.7. Applying the estimates for the coefficients given by (1.9), and making use of the relations (3.6) and |(a)n| ≤ (|a|)n, we get

n=2n(1+nλ-λ-αn)|(a)n-1(b)n-1(c)n-1(1)n-1an|n=2n[n(λ-α)+(1-λ)](a)n-1(b)n-1(P1)n-1(c)n-1(1)n-1(1)n-1=n=2(n-1)[(n-1)(λ-α)+(1-α)](a)n-1(b)n-1(P1)n-1(c)n-1(1)n-1(1)n-1+n=2[(n-1)(λ-α)+(1-α)](a)n-1(b)n-1(P1)n-1(c)n-1(1)n-1(1)n-1=(λ-α)n=2(a)n-1(b)n-1(P1)n-1(c)n-1(1)n-2(1)n-2+(1-α)n=2(a)n-1(b)n-1(P1)n-1(c)n-1(1)n-1(1)n-2+(λ-α)n=2(a)n-1(b)n-1(P1)n-1(c)n-1(1)n-1(1)n-2+(1-α)n=2(a)n-1(b)n-1(P1)n-1(c)n-1(1)n-1(1)n-1=(λ-α)n=2(a)n-1(b)n-1(P1)n-1(c)n-1(1)n-2(1)n-2+(1+λ-2α)n=2(a)n-1(b)n-1(P1)n-1(c)n-1(1)n-1(1)n-2+(1-α)n=2(a)n-1(b)n-1(P1)n-1(c)n-1(1)n-1(1)n-1=(λ-α)abP1cF32(1+a,1+b,1+P1;1+c,1;1)+(1+λ-2α)abP1cF32(1+a,1+b,1+P1;1+c,2;1)+(1-α)[F32(a,b,P1;c,1;1)-1]1-α

provided the hypothesis is satisfied.

We state the following theorems without proof.

### Theorem 3.12

• (i) If a, b > −1, c>0 and ab < 0, then zF(a, b, c) is in G(λ, α) if and only if c>a+b+1-(λ-α)(1-α)ab .

• (ii) If a, b > 0, c > a+b+1, then F1(a, b, c; z) = z[2−F(a, b, c; z)] is in G(λ, α) iffΓ(c)Γ(c-a-b)Γ(c-a)Γ(c-b)[1+(λ-α)ab(1-α)(c-a-b-1)]2.

### Theorem 3.13

• (i) If a, b > −1, c > 0 and ab < 0, then zF(a, b, c) is in M(λ, α) if and only if (λ−α)(a)2(b)2 +(1−4α+3λ)ab(cab−2)+(1 − α)(cab − 2)2 ≥ 0.

• (ii) If a, b > 0, c>a+b+2, then F1(a, b, c; z) = z[2−F(a, b, c; z)] is in M(λ, α) iffΓ(c)Γ(c-a-b)Γ(c-a)Γ(c-b)[1+(λ-α)(a)2(b)2(1-α)(c-a-b-2)2+1-4α+3λ1-αabc-a-b-1]2.

### References

1. de Branges, L (1985). A proof of the Bieberbach conjecture. Acta Math. 154, 137-152.
2. Carlson, BC, and Shaffer, DB (1984). Starlike and prestarlike hypergeometric functions. SIAM J Math Anal. 15, 737-745.
3. Caplinger, TR, and Causey, WM (1973). A class of univalent functions. Proc Amer Math Soc. 39, 357-361.
4. Dixit, KK, and Pal, SK (1995). On a class of univalent functions related to complex order. Indian J Pure Appl Math. 26, 889-896.
5. Dziok, J, and Srivastava, HM (1999). Classes of analytic functions associated with the generalized hypergeometric function. Appl Math Comput. 103, 1-13.
6. Dziok, J, and Srivastava, HM (2003). Certain subclasses of analytic functions associated with the generalized hypergeometric function. Intergral Transforms Spec Funct. 14, 7-18.
7. Goodman, AW (1991). On uniformly convex functions. Ann Polon Math. 56, 87-92.
8. Hohlov, YE (1978). Operators and operations in the class of univalent functions. Izv Vysš Učebn Zaved Matematika. 10, 83-89.
9. Kiryakova, V (2011). Criteria for univalence of the Dziok-Srivastava and the Srivastava-Wright operators in the class A. Appl Math Comput. 218, 883-892.
10. Kanas, S, and Srivastava, HM (2000). Linear operators associated with k-uniformly convex functions. Integral Transform Spec Funct. 9, 121-132.
11. Kanas, S, and Wiśniowska, A (1999). Conic regions and k-uniform convexity. J Comput Appl Math. 105, 327-336.
12. Kanas, S, and Wiśniowska, A (2000). Conic regions and k-starlike functions. Rev Roumaine Math Pures Appl. 45, 647-657.
13. Padmanabhan, KS (1970). On a certain class of functions whose derivatives have a positive real part in the unit disc. Ann Polon Math. 23, 73-81.
14. Ponnusamy, S, and R⊘nning, F (1997). Duality for Hadamard products applied to certain integral transforms. Complex Variables Theory Appl. 32, 263-287.
15. Srivastava, HM (2007). Some Fox-Wright generalized hypergeometric functions and associated families of convolution operators. Appl Anal Discrete Math. 1, 56-71.
16. Srivastava, HM, Murugusundaramoorthy, G, and Sivasubramanian, S (2007). Hypergeometric functions in the parabolic starlike and uniformly convex domains. Integral Transform Spec Funct. 18, 511-520.