Scientific Bulletin of the “Petru Maior”8QLYHUVLW\RI7vUJX0XUHú Vol. 13 (XXX) no. 1, 2016
ISSN-L 1841-9267 (Print), ISSN 2285-438X (Online), ISSN 2286-3184 (CD-ROM)
ABOUT A GENERALIZED CLASS OF CLOSE-TO-CONVEX
FUNCTIONS DEFINED BY THE q-DIFFERENCE OPERATOR
Olga ENGEL, Cosmina NAICU
Babe¸s - Bolyai University
Mihail Kog˘alniceanu Street
,
no.1
,
400084
,
Cluj-Napoca
,
Romania
engel_olga@hotmail.com, cosmina_naicu@yahoo.com
Abstract
In this paper we generalize the class of close-to-convex functions by theq-difference operator, for functions with negative coefficients and we study some properties of this generalized class. An analogue of the Pólya-Schoenberg conjecture is proved.
Keywords: close-to-convex functions, q-derivative, Pólya-Schoenberg conjecture
1
Introduction
LetU = {z ∈ C : |z| < 1} be the unite disk in the complex planeC.We denote byAthe class of the functionsf, normalized byf(0) = 0 = f′
(0)−1, which are analytic inU.
We say that f is starlike in U if f : U → Cis univalent and f(U)is a starlike domain with respect to origin. It is well-known thatf ∈ Ais starlike inU
if and only if
Rezf
′
(z)
f(z)
>0, for allz∈U.
We denote byS∗
the class of starlike functions. LetKbe the class of convex functions. We say that
f ∈ Ais convex inU,iff : U →Cis univalent and
f(U)is a convex domain inC. It is known that the functionf ∈ Ais convex inU if and only if
Re1 +zf
′′
(z)
f′( z)
>0.
We say thatf ∈ Ais close-to-convex inU, if there is a convex functiong∈ Kfor which
Ref
′
(z)
g′(z) >0, z∈U.
Since ifg∈ Kthenzg′
∈S∗
, an equivalent defini-tion of close-to-conexity is the following.
The functionf ∈ Ais close-to-convex inUif there exist a starlike functiong∈S∗
for which
Rezf
′
(z)
g(z) >0, z∈U.
We denote byCthe class of close-to-convex func-tions.
LetT denote a subclass ofA consisting of func-tionsf of the form
f(z) =z−
∞
X
j=2
ajzj, (1)
whereaj ≥ 0, j = 2,3, ... andz ∈ U. A function
f ∈Tis called a function with negative coefficients. If
f ∈Tandfis univalent, the followings are equivalent [9]:
(i)
∞
P
j=2
jaj≤1,
(ii) f ∈T,
(iii) f ∈T∗
,whereT∗
=T∩S∗
.
In our paper we generalize the class of close-to-convex functions, for functions with negative coeffi-cients, and we obtain some intresting results on this generalized class.
In 1973, Ruscheweyh and Sheil-Small [7] proved the Pólya-Schoenberg conjecture, namely ifφis con-vex andf ∈S∗
orK, thenf ∗φalso belong toS∗
or
K. In our paper we prove the analogue of this conjec-ture for the generalized class of close-to-convex func-tions.
c
2
Preliminaries
To prove our main results we need the following preliminary definitions and theorems.
Let
f(z) =z−
∞
X
j=2 ajzj
and
g(z) =z−
∞
X
j=2 bjzj,
then the Hadamard product or the convolutions of the functionsf andgis defined by
(f∗g)(z) =z−
∞
X
j=2
ajbjzj= (g∗f)(z).
In 1908 Jackson introduced the Euler-Jackson q-difference oprator.
Forf ∈ A of the form (1) and0 < q < 1, the q-derivative of the functionf is defined by
Dqf(z) =f(qz)−f(z)
(q−1)z , (2)
wherez6= 0andDqf(0) =f′
(0).
From(2)we can deduce that
Dqf(z) = 1−
∞
X
j=2
1−qj
1−qajz j−1
,
wherez6= 0.
We note with [z0, z1, ..., zn;f] the divided
differ-ences at a system of distinct pointsz0, z1, ..., zn, where
[z0, z1, ..., zn;f] = n
X
j=0
f(zj)
(zj−z0)·...·(zj−zn) .
Then-th orderqderivative of the functionf is de-fined in [8] as follows:
(Dnqf)(z) = [n]q[z, qz, ..., qnz;f], (3)
where[n]q= qn−1
q−1 .
The q-derivative have several intresting application in quantum mechanics and generally in physics.
Theorem 2.1. [10] Let f : Uq ⊆ C → C be q-derivable of order n, then
(Dqnf)(z) = (q−1) −n
z−n q−Cn2·
·
n
X
j=0
[n]q!
[j]q![n−j]q!
(−1)jqCj2f(qn−jz).
A generalization of the class of close-to-convex functions by Dq difference operator we can found in [8]. In followings are given some generalizations of the class of close-to-convex functions, in the case of functions with negative coefficients, using the q -difference operator.
Definition 2.1. A functionf ∈T is said to be in the generalized class of close-to-convex functions of order γ, denoted byU CCq(γ), if
RezDqf(z)
g(z) ≥γ,
where0≤γ <1andg∈T∗ .
Remark 2.1. Ifγ= 0thenU CCq(0) =U CCq.
Definition 2.2. A functionf ∈T is said to be in the generalized class of close-to-convex functions of or-derγ, relative to a fixed functiong ∈T∗
, denoted by U CCq(g, γ), if
RezDqf(z)
g(z) ≥γ,
where0≤γ <1.
Definition 2.3. A functionf ∈T is said to be in the classU CCqn, if
RezD
n qf(z) g(z) >0,
whereg∈T∗ .
3
Main results
Theorem 3.1. Letf(z) =z−
∞
P
j=2
ajzj,aj ≥0,j∈
{2,3, ...}and0≤γ <1.
If f ∈ U CCq(γ), then there exist g ∈ T∗
, g(z) =
z−
∞
P
j=2
bjzj, such that
∞
X
j=2
1−qj
1−qaj−γbj
<1−γ. (4)
If
∞
X
j=2
1−qj
1−qaj<1−γ, (5)
thenf ∈U CCq(γ). In the particular case when
1−qj
1−qaj≤bj, j∈ {2,3, ...}
the inequality (4) is necessary and sufficient condition forf to belongs toU CCq(γ).
Proof. Letf ∈ U CCq(γ), then there existsg ∈ T∗
,
g(z) =z−
∞
X
j=2
bjzjsuch that
RezDqf(z)
Ifz∈[0,1), we obtain
z−
∞
P
j=2
1−qj
1−qajz j z− ∞ P j=2 bjzj
> γ. (6)
We note that forz∈[0,1)we havez−
∞
X
j=2
bjzj >0,
because g ∈ T∗
and in this case
∞
X
j=2
jbj ≤ 1, then
∞
X
j=2 bj <1.
The relation (6) is equivalent to
∞
X
j=2
1−qj
1−qaj−γbj
zj−1
<1−γ.
For (5) we choseg(z) =z. Then
γ−RezDqf(z)
g(z) −1
<1
is true if
γ+|Dqf(z)−1|<1,
but we have
γ+|Dqf(z)−1| ≤
∞ X j=2
1−qj
1−q aj
+γ = ∞ X j=2
1−qj
1−qaj+γ.
To prove the particular case, we suppose
1−qj
1−qaj≥bj, j∈ {2,3, ...}.
Then we have
γ+ ∞ X j=2 bj−
1−qj
1−qaj
|z|
j−1
1−
∞
X
j=2
bj|z|j−1
≤γ+
∞
X
j=2
1−qj
1−qaj−bj
1− ∞ X j=2 bj = γ+ ∞ X j=2
1−qj
1−qaj−γbj
1− ∞ X j=2 bj
<1,
if we suppose that the inequality (4) is true.
Theorem 3.2. Letf(z) =z−
∞
X
j=2
ajzj, g(z) =z−
∞
X
j=2
bjzj, g∈T∗
, whereaj, bj ≥0,j∈ {2,3, ..}and
γ∈[0,1).
Iff ∈U CCq(g, γ), then
∞
X
j=2
1−qj
1−qaj−γbj
<1−γ. (7)
If
∞
X
j=2
h1−qj
1−qaj+ (2−γ)bj
i
<1−γ, (8)
thenf ∈U CCq(g, γ). In the particular case when
1−qj
1−qaj≤bj, j∈ {2,3, ..},
then (7) implies thatf ∈U CCq(g, γ).
Remark 3.1. When f2(z) = z − z2
1 +q ∈
U CCq(g2, γ), whereg2(z) =z− z2
2 ∈T
∗
we have
RezDqf2(z)
g2(z)
= Re z(1−z)
z 1−z 2
= 2 Re
1−z
2−z >0.
But
∞
X
j=2
1−qj
1−qaj+ (2−γ)bj= 1 +
2−γ
2 = 2−
γ
2 ≮1.
This show that (8) is only a sufficies condition.
Because in [7] the authors proved that the convolu-tion of a starlike and a convex funcconvolu-tion is starlike, we can give the following theorem.
Theorem 3.3. Letf(z) =z−
∞
X
j=2
ajzj,g(z) = z−
∞
X
j=2
bjzj and let φ(z) = z−
∞
X
j=2
cjzj convex in U,
whereaj, bj, cj ≥0,j∈ {2,3, ..}.
If f ∈ U CCq(g, γ), where 1−q j
1−qaj ≤ bj for j ∈
{2,3, ..}, thenf ∗φ∈U CCq(g, γ).
Proof. Let
(f∗φ)(z) =z−
∞
X
j=2 ajcjzj.
We know from Definition 2.2 that if (f ∗φ)(z) ∈
U CCq(g, γ), then
where(g∗φ)∈T∗
and0≤γ <1.
Supposef ∈ U CCq(g, γ). Then by Theorem 3.2 we have
∞
X
j=2
h1−qj
1−qaj−γbj
i
<1−γ. (9)
To finish our proof, we must to show
∞
X
j=2
h1−qj
1−qajcj−γbjcj
i
<1−γ.
Sinceφ∈T, the above inequality is equivalent to
∞
X
j=2
|cj|h1−q
j
1−q aj−γbj
i
<1−γ. (10)
Because φis convex, by the coefficient delimitation theorem for convex functions we have |cj| ≤ 1, for
j= 2,3....
Then from (10) we obtain
∞
X
j=2
|cj|h1−q
j
1−qaj−γbj
i
< ∞
X
j=2
h1−qj
1−qaj−γbj
i
<1−γ,
and the proof is done.
Theorem 3.4. Letf(z) =z−
∞
X
j=2
ajzj,g(z) =z−
∞
X
j=2
bjzj ∈T∗ and
F(z) =Icf(z) = c+ 1
zc z
Z
0
f(t)tc−1
dt, c∈N∗ .
If f ∈ U CCq(g, γ), where 1−q j
1−qaj ≤ bj for j ∈
{2,3, ..}, thenF ∈U CCq(g, γ).
Proof. Supposef ∈U CCq(g, γ). Then by Theorem 3.2 we have
∞
X
j=2
h1−qj
1−qaj−γbj
i
<1−γ.
We know that iff has the form (1) then
F(z) =c+ 1
zc z
Z
0
f(t)tc−1 dt=
z−
∞
X
j=2 c+ 1
c+jajz j=
z−
∞
X
j=2 cjzj.
If the
∞
X
j=2
h1−qj
1−q · c+ 1
c+jaj−γbj
i
<1−γ
inequality is true, then according to Theorem 3.2 we know thatF ∈U CCq(g, γ).
Next we prove
1−qj
1−q · c+ 1
c+jaj−γbj
<1−q j
1−qaj−γbj, forj ≥2.
(11)
The inequality (11) is equivalent to
1−qj
1−q aj
1−c+ 1
c+j
>0,
which is true for allc∈N∗
andj ≥2, and the proof is done.
4
Conclusion
The class of close-to-convex functions has an im-portant role in geometric function theory and it was introduced by W. Kaplan.
In the Definition 2.1 we have generalized the class of close-to-convex functions, using the q-difference operator. In the Definition 2.2 we have generalized the class of close-to-convex functions relative to a fixed functiong∈T∗
.
For this two generalized class of functions, in The-orem 3.1 and TheThe-orem 3.2 we have proved several co-efficient inequalities. Using these inequalities we have proved an analogue of the Pólya-Schoenberg conjec-ture, and finally we have showed the preserving prop-ertie of the Bernardi integral operator defined on the
U CCq(g, γ)class.
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