R E S E A R C H
Open Access
Properties for certain subclasses of analytic
functions with nonzero coefficients
Hyo Jeong Lee
1, Nak Eun Cho
1*and Shigeyoshi Owa
2*Correspondence:
[email protected] 1Department of Applied
Mathematics, Pukyong National University, Busan, 608-737, Korea Full list of author information is available at the end of the article
Abstract
In the present paper, we obtain some mapping and inclusion properties for subclasses of analytic functions by using a linear operator defined by the Gaussian hypergeometric function.
MSC: 30C45
Keywords: univalent function; uniformly starlike function; uniformly convex function; Gaussian hypergeometric function; Hadamard product
1 Introduction
LetAdenote the class of functions of the form
f(z) =z+
∞
n=
anzn (an= ), (.)
which are analytic in the open unit diskU={z∈C:|z|< }. We also denote bySthe class of all functions inAwhich are univalent inU.
A functionf ∈Ais said to be in the classt(A,B,α) if
t(A–Bf) –(z) – B(f(z) – )
<α (z∈U), (.)
whereAandBare complex numbers withA=B,t∈C\{}, andαis a positive real number. In particular, for some real numbersAandBwith –≤B<A≤ andα= without any restriction of the coefficientsan(n∈N={, , . . .}) the classt(A,B,α) was introduced
by Dixit and Pal []. Moreover, by giving specific valuest,A,B, andαin (.), we obtain subclasses studied by various researchers in earlier works (see [–]).
A functionf ∈Ais said to be in the classUST(α) if
f(z) –f(ξ) (z–ξ)f(z)
>α (z×ξ∈U×U; ≤α< ).
Furthermore, a functionf∈Ais said to be in the classUCV(α) if
+(z–ξ)f
(z)
f(z)
>α (z×ξ∈U×U; ≤α< ).
The classes UST()≡UST andUCV()≡UCV are introduced by Goodman [, ] (they are called the classes of uniformly starlike and uniformly convex functions, respec-tively). The classes of uniformly starlike and uniformly convex functions have been exten-sively studied by Ma and Minda [] and Rønning [].
Let us consider the Gaussian hypergeometric functionF(a,b;c;z) defined by
F(a,b;c;z) =
∞
n=
(a)n(b)n
(c)n
zn (a,b,c∈C;c= , –, –, . . . ;z∈U),
where (ν)nis the Pochhammer symbol (or the shifted factorial) defined (in terms of the
gamma function) by
(ν)n: =
(ν+n) (ν)
=
ifn= andν∈C\ {},
ν(ν+ )· · ·(ν+n– ) ifn∈Nandν∈C. We note thatF(a,b;c;z) =F(b,a;c;z) and
F(a,b;c; ) =(c–a–b)(c) (c–a)(c–b)
{c–a–b}> .
We also recall (see [, ]) that the functionF(a,b;c;z) is bounded if{c–a–b}> , and it has a pole atz= if{c–a–b} ≤. Moreover, univalence, starlikeness and convexity properties ofzF(a,b;c;z) have been extensively studied by Ponnusamy and Vuorinen [] and Ruscheweyh and Singh [].
Forf ∈A, we define the operatorIa,b;cf by
Ia,b;cf(z) =zF(a,b;c;z)∗f(z), (.)
where∗denotes the usual Hadamard product (or convolution) of power series. For a spe-cial case of the operator Ia,b;cf, we can refer to the paper by Swaminathan [] and the
references cited therein.
In this paper, we obtain a necessary condition for the class t(A,B,α) and sufficient
conditions for the classest(A,B,α),UST(α), andUCV(α), respectively. Moreover, we
find a condition for univalency of the operatorIa,b;cf defined by (.). 2 Main result
Theorem Let a function f of the form(.)be in the classt(A,B,α)with a
n=|an|ei (n+)π
(n∈N\ {}),then
∞
n=
n –α|B||an| ≤α|t||A–B|. (.)
Proof From the definition oft(A,B,α), we have
and so
∞
n=
nanzn–
<α
t(A–B) –B
∞
n=
nanzn–
. (.)
If we takez=reiπ, then we see that
anzn–=|an|rn– (≤r< ). (.)
Then, by using (.) to (.), we have
∞
n=
n|an|rn–<α
t(A–B) –B
∞
n=
n|an|rn–
<αt(A–B)+α|B|
∞
n=
n|an|rn–,
or, equivalently,
∞
n=
n –α|B||an|rn–<α|t||A–B|. (.)
Lettingr→– in (.), we have the inequality (.). Thus we complete the proof of
Theorem .
Theorem Let a function f of the form(.)be in the classA.If
∞
n=
n +α|B||an| ≤α|t||A–B|, (.)
where A and B are complex numbers with A=B,t∈C\{},andαis a positive real number,
then f ∈ t(A,B,α).The result is sharp for the function defined by
f(z) =z+
∞
n=
αt(A–B)
n(n– )( +α|B|)z n
A,B∈C;A=B;t∈C\ {};||= ;z∈U.
Proof In view of the definition oft(A,B,α), it suffices to prove that
f(z) – <αt(A–B) –Bf(z) – (z∈U). (.)
From (.), we have
∞
n=
nanzn–
<α
t(A–B) –B
∞
n=
nanzn–
Hence it is sufficient to show that
∞
n=
n|an|rn–<α |t||A–B|–|B| ∞
n=
n|an|rn–
,
which is equivalent to the relation
∞
n=
n +α|B||an|rn–<α|t||A–B|. (.)
Lettingr→–in (.), we have ∞
n=
n +α|B||an| ≤α|t||A–B|.
Therefore, by the assumption (.), we prove thatf∈ t(A,B,α).
Theorem Let a function f of the form(.)be in the classA.If
∞
n=
( –α)n– |an| ≤ –α (≤α< ), (.)
then f ∈UST(α).The result is sharp for the function
f(z) =z+
∞
n=
( –α)
n(n– )(( –α)n– )z
n ≤α< ;||= .
Proof It suffices to show that
(fz(z–) –ξ)ff(ξ(z))–
≤ –α ≤α< ; (z,ξ)∈U×U.
We have
(fz(z–) –ξ)ff(ξ(z))–
=
∞
n=an(ξn–+zξn–+· · ·+zn–) –
∞
n=nanzn–
+∞n=nanzn–
≤
∞
n=(n– )|an|
–∞n=n|an|
,
which is bounded by –αif the assumption (.) is satisfied. Thus we complete the proof
of Theorem .
Theorem Let a function f of the form(.)be in the classA.If
∞
n=
then f ∈UCV(α).The result is sharp for the function
f(z) =z+
∞
n=
( –α)
n(n– )(n– –α)z
n ≤α< ;||= ;z∈U.
Proof It suffices to show that
(z–fϕ)(zf)(z)
< –α ≤α< ; (z,ϕ)∈U×U.
We have
(z–fϕ)(zf)(z)
=(z–ϕ)
∞
n=n(n– )anzn–
–∞n=nanzn–
≤
∞
n=n(n– )|an|
–∞n=n|an|
,
which is bounded by –αif the assumption (.) is satisfied. Thus we complete the proof
of Theorem .
Theorem Let a,b∈C\ {}and c>|a|+|b|.If f ∈ t(A,B,α)with a
n=|an|ei (n+)π
,
<|B|< ,and
(c–|a|–|b|)(c) (c–|a|)(c–|b|)≤
–α|B|
+α|B|+ ,
then Ia,b;cf ∈ t(A,B,α),where the operator Ia,b;cf is defined by(.).
Proof We want to prove from Theorem that
T:= ∞
n=
n +α|B||An| ≤α|t||A–B|,
where
An=
(a)n–(b)n–
(c)n–()n–
an.
Since, from Theorem ,
|an| ≤
α|t||A–B| n( –α|B|), T≤
α|t||A–B|( +α|B|)
–α|B|
∞
n=
(|a|)n–(|b|)n–
(c)n–()n–
=α|t||A–B|( +α|B|) –α|B|
∞
n=
(|a|)n(|b|)n
(c)n()n
–
=α|t||A–B|( +α|B|) –α|B|
(c–|a|–|b|)(c) (c–|a|)(c–|b|)–
≤α|t||A–B|,
We introduce the following lemma which is needed for the proof of the next theorem.
Lemma [] Letωbe regular in the unit diskUwithω() = .Then,if|ω(z)|attains a maximum value on the circle|z|=r(≤r< )at a point z,we can write
zω(z) =kω(z) (k≥).
Theorem Let a function f of the form(.)be in the classA.Assume
(Ia,b;cf(z))–
–α
β z(Ia,b;cf(z))
(Ia,b;cf(z))–α
γ<
γ (z∈U) (.)
for some realα(≤α< ),β> ,andγ > .Then
Ia,b;cf(z)
– < –α (z∈U). (.)
Proof Let us defineωby
ω(z) = (Ia,b;cf(z))
–
–α (z∈U).
Then it follows thatωis analytic inUwithω() = . By (.), we have
ω(z)β zω
(z)
ω(z) +
γ =ω(z)β+γzω
(z)
ω(z) ω(z) +
γ
<
γ (z∈U). (.)
Suppose that there exists a pointz∈Usuch that max
|z|≤|z|
ω(z)=ω(z)= .
Then, by Lemma , we can put
zω(z)
ω(z)
=k≥.
Hence, we obtain
ω(z)
β
zω(z)
ω(z) +
γ=zω
(z )
ω(z) +
γ
≥
k
γ
≥
γ,
which contradicts the condition (.). This shows that
ω(z)=(Ia,b;cf(z))
–
–α
< (z∈U).
Remark The condition (.) in Theorem implies that
Ia,b;cf(z)
> (z∈U).
Therefore, by the Noshiro-Warschawski theorem [], the operatorIa,b;cf is univalent in Uunder the restrictions of Theorem .
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors jointly worked on the results and they read and approved the final manuscript.
Author details
1Department of Applied Mathematics, Pukyong National University, Busan, 608-737, Korea.2Department of Mathematics,
Faculty of Education, Yamato University, Katayama 2-5-1, Suita, Osaka 564-0082, Japan.
Acknowledgements
The authors would like to express their thanks to the referees for many valuable advices regarding a previous version of this paper. This research was supported for the second author by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2011-0007037).
Received: 18 July 2014 Accepted: 2 December 2014 Published:12 Dec 2014 References
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