A Family of Convolution Operators for Multivalent Analytic Functions

Vol. 5, No. 4, 2012, 469-479

ISSN 1307-5543 – www.ejpam.com

A Family of Convolution Operators for Multivalent Analytic

Functions

Janusz Sokół

_{, Khalida Inayat Noor}

_{, Hari Mohan Srivastava}

1 _{Department of Mathematics, Rzeszów University of Technology, Al. Powsta´nców Warszawy 12,}

35-959 Rzeszów, Poland

2_{Department of Mathematics, COMSATS Institute of Information Technology, Park Road,}

Islamabad, Pakistan

3_{Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W}

3R4, Canada

Abstract. In this paper, we consider a family of multiplier transformations and several subclasses of multivalent functions which are defined by means of convolution. Several interesting results are derived. Some (known or new) special cases of the multivalent function classes, which are investigated here, are also discussed.

2010 Mathematics Subject Classifications: 30C45; 30C80, 33C05, 33C20

Key Words and Phrases: Univalent functions, Convex functions, Starlike functions, Subordination be-tween analytic functions, Hadamard product (or convolution), Gauss and generalized hypergeometric functions

1. Introduction and Definitions

LetA(p)denote the class of functions of the following form:

f(z) =zp+

∞

X

n=1

a_p+nzp+n (p∈N:={1, 2, 3, . . .}), (1)

which are analytic in the open unit disk

U={z:z∈C and |z|<1}.

∗_{Corresponding author.}

Email addresses:jsokolprz.edu.pl(J. Sokół),khalidanoorhotmail.om(K. Noor),

harimsrimath.uvi.a(H. M. Srivastava)

Let f,g∈ A(p), f be given by (1) and

g(z) =zp+

∞

X

n=1

b_p+nzp+n. (2)

Then the Hadamard product (or convolution) of f andgis defined by

(f ∗g)(z):=zp+

∞

X

n=1

ap+nbp+nzp+n=:(g∗f)(z) (p∈N). (3)

Also, if f and gare analytic inU, we say that f is subordinate to ginU, and we write

f ≺g (z∈U), (4)

if there exists a Schwarz functionw such that f(z) = g w(z)

and _|w(z)_{| ≤ |}z_| (z_∈U).

We now define a linear operator Lc_k:A(p)→ A(p)as follows: let the linear operator L₀ L₀:A(p)→ A(p) (k∈N; c∈C\ {0}) (5) be given and

c L_kcf(z) =z L_kc₋₁f(z)′

+ (c₋p)Lc_k−1f(z) (6) with

L₀c:=L₀. (7)

It can easily be seen from (6) that the operator L_kc is linear and it satisfies the following property:

L₀cf(z) =zp+

∞

X

n=1

Ap+nzp+n, (8)

which implies that

Lc_kf(z) =zp+

∞

X

n=1

(1+n/c)kA_p+nzp+n. (9)

We also have

c L₁cf =z(Lc₀f)′+ (c−p)Lc₀f, (10) c Lc_kf =zp+1(z−pLc_k−1f)′+c Lc_k−1f (11) and

L_kcf zp =

z c

‚ Lc_k−1f

zp

Œ′ + L

c k−1f

zp . (12)

By appropriately choosing Lc_kgiven by (8), we obtain several applications studied by var-ious earlier authors (see, for example, [2, 3, 4, 5, 6, 8, 14, 9, 10, 11, 15, 20]; see also

Definition 1. Letqand h be analytic inU. Also let the function h be convex univalent inUwith

h(0) =q(0) =1. Thenq∈ P(h)if and only if

q(z)≺h(z) (z∈U). (13)

Some well-known examples of the convex functionhare listed below. (i) If

h(z) = 1+ (1−2α)z

1−z and 0≦α <1, then

ℜ h(z)

> α (z∈U; 0≦α <1).

(ii) If

h(0) =1 and h(z) =

_1+z 1−z

β

(0< β <1),

then

arg h(z)<

βπ

2 (z∈U). (iii) Let

h(z) = M(1+z) M+ (1₋M)z

M> 1

2

. Also

h(U) =_{w:|w−M|<M}. (iv) If

h(z) =pz+1 and ℜpz+1

≧0 (z∈U), thenh(U)is the interior of the right part of the Bernoulli lemniscate[see 1]. (v) If

h(z) =1+ 2

π2 –

log ‚

1+pz 1−pz

Ϊ2

and ℑ€pzŠ>0 (z∈U),

thenh(U)is the interior of the parabola given by ¦

w:[ℑ(w)]2=2ℜ(w)−1©.

Definition 2. Let L₀ be a linear operator onA(p)and let L_kc be given by(6). Then, forλ≧0, a function f ∈ A(p)is said to be in the classS_kc(p,λ;h)if and only if

‚

(1₋λ)L

c kf(z)

zp +λ

Lc_k+1f(z) zp

Œ

2. Preliminary Results

We need each of the following lemmas in our present investigation.

Lemma 1(see[7]and[12]). Let h be an analytic and convex univalent function inU. Let the function f be analytic inUwith h(0) = f(0) =1. If

f(z) +z f

′_(z)

γ ≺h(z) z∈U; ℜ(γ)≧0; γ6=0

, (15)

then

f(z)_≺g(z) = γ zγ

Z z

0

tγ−1h(t)dt ≺h(z) (z∈U).

Moreover, the function g is convex univalent inUand it is the best dominant of the subordination (15)in the sense that in the sense that f ≺ g for all f satisfying(15), and if there exists q such that f ≺q for all f satisfying(15), then g≺q.

Lemma 2 (see[19]). Let the functions qand h be analytic in U withq(0) =1. Suppose also

that

ℜ q(z)>

1

2 (z∈U). Then

(q∗h)(U)_⊂co{h(U)_}, whereco{h(U)_}is the convex hull of h(U).

Lemma 3(see[16]). Let

f(z)≺F(z) (z∈U) and g(z)≺G(z) (z∈U).

If the functions F and G are convex inU, then

(f ∗g)(z)_≺(F∗G)(z) (z∈U).

Unless otherwise stated, we shall assume throughout this paper that

λ≧0, c∈C\ {0}, ℜ(c)>0, k,p∈N, and z∈U.

3. Main Results

Our first main result in this paper is contained in Theorem 1 below. Theorem 1. If the function f belongs to the classS_kc(p,λ;h), then

L_kcf(z)

Moreover, ifλ >0,then

L_kcf(z)

zp ∈ P(g), (16)

where

g(z) = c

λz

−c

λ

Z z

0

tλc−1h(t)dt≺h(z) (z∈U), (17)

the function g is convex univalent inUand g is the best dominant of the subordination

Lc_kf(z)

zp ≺ g (z∈U).

Proof. The proof for the case whenλ=0 is trivial. We, therefore, suppose thatλ >0. Let

f ∈ S_kc(p,λ;h). (18)

Then, by (12), we have

(1₋λ)L

c kf(z)

zp +λ

L_kc₊₁f(z)

zp =

Lc_kf(z) zp +

λz c

‚ Lc_kf(z)

zp

Œ′

∈ P(h). (19)

Let the functionH(z)be given by

H(z):= L

c kf(z)

zp (z∈U). (20)

Then, by (19), it follows that

H(z) +λ czH

′_{(z)∈ P(h)}

and

H(z) +λ czH

′_{(z)≺h(z) (z∈U). (21)}

Now, using Lemma 1 in (21) with

γ= c

λ and λ >0, (22)

we obtain (17). This shows that H ∈ P(g), where the function g is given by (17). Conse-quently, the proof of Theorem 1 is complete.

We take

L0f(z) = f(z)∗φ(a,c,z), (23) where

φ(a,c,z) =

∞

X

n=0 (a)_n (c)_nz

and(λ)_n is the Pochhammer symbol defined, in terms of the familiar Gamma function, by

(λ)_n=Γ(λ+n) Γ(λ) =

(

1 (n=0; λ6=0),

λ(λ+1). . .(λ+n−1) (n∈N),

it being understoodconventionallythat(0)₀:=1. We also let

h(z) = 1+Az

1+Bz (−1≦B<A≦1). (24)

Then, by applying Theorem 1, we obtain the subordination (17) with

g(z) =    A B+ 1− A

B

(1+Bz)−₂1F1

1, 1;c−1

λ +1;

Bz Bz+1

(B6=0)′

1−

c−1 c−1+λ

Az (B=0),

where₂F₁is the Gauss hypergeometric function defined by

2F1(α,β;γ;z):=

∞

X

n=0

(α)_n(β)_n (γ)_n

zn

n! (z∈U; γ=6 0,−1,−2,−3, . . .). (25)

Theorem 2. Let0≦λ1≦λ2. Then

Sc

k(p,λ2;h)⊂ S c

k(p,λ1;h). (26)

Proof. Suppose that f ∈ S_kc(p,λ2;h). A simple computation will then yield

(1₋λ1) Lc_kf(z)

zp +λ1

Lc_k+1f(z) zp

=

1₋λ1

λ2 _Lc

kf(z)

zp +

λ1

λ2 ‚

(1₋λ2) L_kcf(z)

zp +λ2

L_kc₊₁f(z) zp

Œ

. (27)

It can now be easily shown that the classP(h)is a convex set. We can write (27) as follows:

(1−λ1) L_kcf(z)

zp +λ1

Lc_k+1f(z)

zp =

1−λ1

λ2

h₁(z) +λ1

λ2

h₂(z) =ψ(z), (28)

where h₁ ∈ P(h), by Theorem 1, andh₂ ∈ P(h), since f ∈ S_kc(p,λ2;h). We thus find that

ψ∈ P(h). Consequently, f ∈ S_kc(p,λ1;h). This proves Theorem 2.

Theorem 3. The following inclusion relationship holds true:

Sc

k(p,λ;h)⊂ S c

Proof. Let f ∈ S_kc(p,λ;h)and suppose that ‚

(1−λ)L

c k−1f(z)

zp +λf r ac L

c kf(z)z

p

Œ

=H(z).

Then, from (12), we have

(1−λ)L

c k−1f(z)

zp +λ

Lc_kf(z) zp

+z

c ‚

(1−λ)L

c k−1f(z)

zp +λ

Lc_kf(z) zp

Œ′

=H(z) +1 c zH

′_(z)

=(1₋λ) 



Lc_k−1f(z)

zp +

z c

‚

Lc_k−1f(z) zp Œ′  +λ   L_kcf(z)

zp +

z c

‚ Lc_kf(z)

zp

Œ′



= ‚

(1₋λ)L

c kf(z)

zp +λ

L_kc₊₁f(z) zp

Œ

∈ P(h).

We thus find that

H(z) +1 c zH

′_{(z)≺h(z) (z∈U). (30)}

By applying Lemma 1, it follows that

H(z)_≺ c zc

Z z

0

tc−1h(t)dt≺h(z) (z∈U),

which shows thatH_{∈ P}(h). Consequently, we have ‚

(1−λ)L

c k−1f(z)

zp +λ

Lc_kf(z) zp

Œ

∈ P(h). (31)

This evidently proves that f ∈ S_kc₋₁(p,λ;h).

Corollary 1. Forℜ(c)>0, let f ∈ S_kc(p,λ;h). Then Lc_sf(z)

zp ∈ P(h) (s∈ {0, 1, 2, . . . ,k}). (32) Proof. We can readily deduce the assertion (32) of the above Corollary from the assertion (17) of Theorem 1. The details involved are being omitted here.

In order to get the convolution results of the multivalent analytic function classS_kc(p,λ;h), it is necessary to put the following restrictions on the operator L_kc:

Theorem 4. Let the operator Lc_ksatisfy the condition(33). If f_j∈ S_kc(p,λ;h_j) (j=1, 2), then each of the following inclusion relationships holds true:

G(z) = (1−λ)Lc_k(f₁∗f₂)(z) +λL_kc₊₁(f₁∗f₂)(z)∈ S_kc(p,λ,h₁∗h₂), (34) L_kc(f₁∗f₂)(z)_{∈ Sk}c(p,λ;h₁∗h₂) (35) and

Lc_k”Lc_k(f₁∗f₂)(z)—

zp ∈ P(h1∗h2). (36)

Proof. Since

f1∈ Skc(p,λ;h1) and f2∈ Skc(p,λ;h2), (37)

it follows that _‚

(1₋λ)L

c kf1(z)

zp +λ

L_kc₊₁f₁(z) zp

Œ

∈ P(h₁) (38)

and

‚

(1₋λ)L

c kf2(z)

zp +λ

L_kc₊₁f₂(z) zp

Œ

∈ P(h2). (39)

Also, from (38), (39) and Theorem 1, we have Lc_kf₁(z)

zp ∈ P(h1) (40)

and

Lc_kf₂(z)

zp ∈ P(h2). (41)

Thus, by making use of (33), (38), (39) and Lemma 3, in conjunction with the technique used before, we have

(1₋λ)L

c k

”

(1₋λ)Lc_k(f1∗f2)(z) +λLkc+1(f1∗f2)(z) —

zp

+λ L

c k+1

”

(1−λ)L_kc(f₁∗f₂)(z) +λLc_k+1(f₁∗f₂)(z)— zp

= ‚

(1−λ)L

c kg(z)

zp +λ

Lc_k+1g(z) zp

Œ

∈ P(h₁∗h₂),

that is,G ∈ S_kc(p,λ;h₁∗h₂). This proves the first assertion (34) of Theorem 4. In order to demonstrate the second assertion (35) of Theorem 4, we again proceed in a similar manner and apply Lemma 3 to (38) and (41). We thus obtain

(1−λ)L

c k

”

Lc_k(f₁∗f₂)(z)—

zp +λ

Lc_k+1”L_kc(f₁∗f₂)(z)— zp

!

which clearly implies (35). Finally, from (42) and Theorem 1, we obtain the third assertion (36) of Theorem 4.

As a special case of Theorem 4, we obtain a result proved in[11](wherec=c1andk=0) for

h_j(z) = 1+Ajz 1+Bjz

(z∈U, j=1, 2) and

Lc_kf(z) = f(z)_{∗ q}F_r(z),

where_qF_r is the generalized hypergeometric function defined by (see also[4]and[5])

qFr(z) = qFr(α1, . . . ,αq;β1, . . . ,βr;z):=

∞

X

n=0

(α1)n. . .(αq)n

(β1)n. . .(βr)n

zn n!

(q,r ∈N₀=N_{∪ {}0_}; q≦r+1) (43) for complex parameters

α1, . . . ,αq and β1, . . . ,βr (βj6=0,−1,−2, . . . ; j=1, . . . ,r). (44)

Theorem 5. Let the operator Lc_k satisfy the condition(33). If f _{∈ Sk}c(p,λ;h) and q∈ A(p)

with

ℜ

q(z)

zp

≧ 1

2 (z∈U), (45)

then f ∗q∈ S_kc(p,λ;h).

Proof. By using the properties of convolution and (33), we have

(1₋λ)L

c

k(f ∗q)(z)

zp +λ

L_kc₊₁(f ∗q)(z)

zp

= ‚

(1−λ)L

c kf(z)

zp +λ

Lc_k+1f(z) zp

Œ ∗q(z)

zp

=H(z)_∗q(z)

zp H∈ P(h)).

Now, by using Lemma 2, we get

H(z)_∗q(z) zp

∈ P(h),

which implies that ‚

(1₋λ)L

c

k(f ∗q)(z)

zp +λ

Lc_k+1(f ∗q)(z)

zp

Œ

∈ P(h). (46)

By means of (46), we have thus proved the assertion of Theorem 5 that

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