Open Access

A class of analytic functions involving in the Dziok-Srivastava operator

Journal of Inequalities and Applications20132013:138

https://doi.org/10.1186/1029-242X-2013-138

Received: 7 December 2012

Accepted: 8 March 2013

Published: 1 April 2013

Abstract

Let A be a class of functions f ( z ) of the form

f ( z ) = z + n = 2 a n z n
(0.1)

which are analytic in the open unit disk U . By means of the Dziok-Srivastava operator, we introduce a new subclass

S m l ( α 1 , α , μ ) ( l m + 1 , l , m N { 0 } , π 2 < α < π 2 , μ > cos α )

of A . In particular, S 0 1 ( 2 , 0 , 0 ) coincides with the class of uniformly convex functions introduced by Goodman. The order of starlikeness and the radius of α-spirallikeness of order β ( β < 1 ) are computed. Inclusion relations and convolution properties for the class S m l ( α 1 , α , μ ) are obtained. A special member of S m l ( α 1 , α , μ ) is also given. The results presented here not only generalize the corresponding known results, but also give rise to several other new results.

MSC:30C45.

Keywords

uniformly convex functionsconvex univalent functionsstarlike functionsα-spirallike functionsconvolutionDziok-Srivastava operatorsubordination

1 Introduction

Let A be a class of functions f ( z ) of the form
f ( z ) = z + n = 2 a n z n
(1.1)
which are analytic in the open unit disk U = { z : | z | < 1 } . For β < 1 , a function f ( z ) A is said to be starlike of order β in U if
z f ( z ) f ( z ) > β ( z U ) .
(1.2)
This class is denoted by S ( β ) ( β < 1 ). For π 2 < α < π 2 and β < 1 , a function f ( z ) A is said to be α-spirallike of order β in U if
{ e i α z f ( z ) f ( z ) } > β cos α ( z U ) .
(1.3)
When 0 β < 1 , it is well known that all the starlike functions of order β and α-spirallike functions of order β are univalent in U . A function f ( z ) A is said to be convex univalent in U if
{ 1 + z f ( z ) f ( z ) } > 0 ( z U ) .
(1.4)
We denote this class by K . Also, let UCV ( K ) be the class of uniformly convex functions in U introduced by Goodman [1]. It was shown in [2] that f ( z ) A is in UCV if and only if
{ 1 + z f ( z ) f ( z ) } > | z f ( z ) f ( z ) | ( z U ) .
(1.5)
In [2], Rønning investigated the class S p defined by
S p = { f ( z ) S ( 0 ) : f ( z ) = z g ( z ) , g ( z ) UCV } .
(1.6)

The uniformly convex and related functions have been studied by many authors (see, e.g., [110] and the references therein).

If
f ( z ) = z + n = 2 a n z n A and g ( z ) = z + n = 2 b n z n A ,
then the Hadamard product (or convolution) of f ( z ) and g ( z ) is given by
( f g ) ( z ) = z + n = 2 a n b n z n .
For
α j C ( j = 1 , 2 , , l ) and β j C { 0 , 1 , 2 , } ( j = 1 , 2 , , m ) ,
the generalized hypergeometric function
F m l ( α 1 , , α l ; β 1 , , β m ; z )
is defined by the following infinite series:
F m l ( α 1 , , α l ; β 1 , , β m ; z ) = n = 0 ( α 1 ) n ( α l ) n ( β 1 ) n ( β m ) n z n n ! ( l m + 1 ; l , m N 0 = N { 0 } ; z U ) ,
where ( c ) n is the Pochhammer symbol defined by
( c ) n = { 1 ( n = 0 ) , c ( c + 1 ) ( c + n 1 ) ( n N ) .
Corresponding to the function
z l F m ( α 1 , , α l ; β 1 , , β m ; z ) ,
the Dziok-Srivastava operator (see [11])
H ( α 1 , , α l ; β 1 , , β m ) : A A
is defined by the following Hadamard product:
H ( α 1 , , α l ; β 1 , , β m ) f ( z ) = ( z l F m ( α 1 , , α l ; β 1 , , β m ; z ) ) f ( z ) ( l m + 1 ; l , m N 0 ; z U ) .
If f ( z ) A is given by (1.1), then we have
H ( α 1 , , α l ; β 1 , , β m ) f ( z ) = z + n = 1 ( α 1 ) n ( α l ) n ( β 1 ) n ( β m ) n a n + 1 n ! z n + 1 ( z U ) .
(1.7)
In order to make the notation simple, we write
H m l ( α 1 ) = H ( α 1 , , α l ; β 1 , , β m ) ( l m + 1 ; l , m N 0 ) .
(1.8)

It should also be remarked that the Dziok-Srivastava operator H m l ( α 1 ) is a generalization of several linear operators considered in earlier investigations (see [1219], also see [20]).

In this paper we introduce and investigate the following subclass of A .

Definition A function f ( z ) A is said to be in S m l ( α 1 , α , μ ) if it satisfies the condition
{ e i α z ( H m l ( α 1 ) f ( z ) ) H m l ( α 1 ) f ( z ) } + μ > | z ( H m l ( α 1 ) f ( z ) ) H m l ( α 1 ) f ( z ) 1 | ( z U ) ,
(1.9)
where
l m + 1 , l , m N 0 , π 2 < α < π 2 and μ > cos α .
(1.10)
Note that f ( z ) = z S m l ( α 1 , α , μ ) and that
S 0 1 ( 1 , α , 0 ) = { f ( z ) A : { e i α z f ( z ) f ( z ) } > | z f ( z ) f ( z ) 1 | ( z U ) } .
(1.11)
Also,
S 0 1 ( 1 , 0 , 0 ) = S p and S 0 1 ( 2 , 0 , 0 ) = UCV .
(1.12)

Throughout this paper we assume, unless otherwise stated, that l, m, α and μ satisfy (1.10).

2 Subordination theorem

Let f ( z ) and g ( z ) be analytic in U . We say that the function f ( z ) is subordinate to g ( z ) in U , and we write f ( z ) g ( z ) , if there exists an analytic function w ( z ) in U such that
| w ( z ) | | z | and f ( z ) = g ( w ( z ) ) ( z U ) .
If g ( z ) is univalent in U , then
f ( z ) g ( z ) f ( 0 ) = g ( 0 ) and f ( U ) g ( U ) .
Theorem 1 A function f ( z ) A is in S m l ( α 1 , α , μ ) if and only if
e i α z ( H m l ( α 1 ) f ( z ) ) H m l ( α 1 ) f ( z ) h ( z ) cos α + i sin α ,
(2.1)
where
h ( z ) = 1 + 2 π 2 ( 1 + μ cos α ) ( log 1 + z 1 z ) 2 = 1 + 8 π 2 ( 1 + μ cos α ) { z + 2 3 z 2 + 23 45 z 3 + } ( z U ) .
(2.2)
Proof Let us define w ( z ) = u + i v by
e i α z ( H m l ( α 1 ) f ( z ) ) H m l ( α 1 ) f ( z ) = w ( z ) cos α + i sin α ( z U ) .
(2.3)
Then w ( 0 ) = 1 and the inequality (1.9) can be rewritten as
u > cos α 2 ( cos α + μ ) v 2 + 1 2 ( 1 μ cos α ) .
(2.4)
Thus
w ( U ) Ω = { w = u + i v : u  and  v  satisfy  ( 2.4 ) } .

It follows from (2.2) that h ( 0 ) = 1 . In order to prove the theorem, it suffices to show that the function w = h ( z ) given by (2.2) maps U conformally onto the parabolic region Ω.

Note that 1 2 ( 1 μ cos α ) < 1 . Consider the transformations
w 1 = w 1 , w 2 = exp ( π w 1 2 cos α cos α + μ ) , t = 1 2 ( w 2 + 1 w 2 ) .
(2.5)
It is easy to verify that the composite function
t = ch ( π 2 cos α ( w 1 ) cos α + μ ) = g ( w ) ( say )
maps Ω + = Ω { w = u + i v : v > 0 } conformally onto the upper half-plane Im ( t ) > 0 so that w = ( w ) [ 1 2 ( 1 μ cos α ) , + ) corresponds to t = ( t ) [ 1 , + ) and w = 1 to t = 1 . With the help of the symmetry principle, the function t = g ( w ) maps Ω conformally onto the region G = { t : | arg ( t + 1 ) | < π } . Since
t = 2 ( 1 + z 1 z ) 2 1
(2.6)
maps U onto G, we see that
w = g 1 ( t ) = 1 + 1 2 π 2 ( 1 + μ cos α ) ( log ( t + t 2 1 ) ) 2 = 1 + 2 π 2 ( 1 + μ cos α ) ( log 1 + z 1 z ) 2 = h ( z )

maps U conformally onto Ω. The proof of the theorem is now completed. □

Corollary 1 Let f ( z ) S m l ( α 1 , α , μ ) . Then for z U ,
| ( H m l ( α 1 ) f ( z ) z ) sec α e i α | exp { 2 π 2 ( 1 + μ cos α ) 0 1 1 ρ ( log 1 + ρ | z | 1 ρ | z | ) 2 d ρ }
(2.7)
and
| ( H m l ( α 1 ) f ( z ) z ) sec α e i α | exp { 8 π 2 ( 1 + μ cos α ) 0 1 1 ρ ( arctan ρ | z | ) 2 d ρ } .
(2.8)

The results are sharp.

Proof

From Theorem 1 we have
e i α cos α ( z ( H m l ( α 1 ) f ( z ) ) H m l ( α 1 ) f ( z ) 1 ) h ( z ) 1
for f ( z ) S m l ( α 1 , α , μ ) and h ( z ) given by (2.2). Since the function h ( z ) 1 is univalent and starlike (with respect to the origin) in U , using the result of Suffridge [[21], Theorem 3], we get
e i α cos α 0 z ( ( H m l ( α 1 ) f ( t ) ) H m l ( α 1 ) f ( t ) 1 t ) d t 0 z h ( t ) 1 t d t .
This implies that
e i α cos α log H m l ( α 1 ) f ( z ) z = 0 1 h ( ρ w ( z ) ) 1 ρ d ρ ( z U ) ,
(2.9)

where w ( z ) is analytic and | w ( z ) | | z | in U .

Noting that h ( z ) maps the disk | z | < ρ ( 0 < ρ 1 ) onto a region which is convex and symmetric with respect to the real axis, we know that
h ( ρ | z | ) { h ( ρ w ( z ) ) } h ( ρ | z | ) ( z U ) .
(2.10)
Now (2.2), (2.9) and (2.10) lead to
log | ( H m l ( α 1 ) f ( z ) z ) sec α e i α | 2 π 2 ( 1 + μ cos α ) 0 1 1 ρ ( log 1 + ρ | z | 1 ρ | z | ) 2 d ρ
and
log | ( H m l ( α 1 ) f ( z ) z ) sec α e i α | 2 π 2 ( 1 + μ cos α ) 0 1 1 ρ ( log 1 + i ρ | z | 1 i ρ | z | ) 2 d ρ = 8 π 2 ( 1 + μ cos α ) 0 1 1 ρ ( arctan ρ | z | ) 2 d ρ

for z U . Hence we have (2.7) and (2.8).

Furthermore, for
α j C { 0 , 1 , 2 , } ( j = 1 , , l ) ,
it is easy to see that the function f 0 ( z ) in S m l ( α 1 , α , μ ) , defined by
H m l ( α 1 ) f 0 ( z ) = z exp { 2 π 2 ( 1 + μ cos α ) cos α e i α 0 z 1 t ( log 1 + t 1 t ) 2 d t } ( z U ) ,
(2.11)

shows that the estimates (2.7) and (2.8) are sharp. □

Corollary 2 Let f ( z ) S m l ( α 1 , α , μ ) , where
α j C { 0 , 1 , 2 , } ( j = 1 , 2 , , l ) .
Then
f ( z ) = z exp { 2 cos α e i α π 2 ( 1 + μ cos α ) 0 1 1 ρ ( log 1 + ρ w ( z ) 1 ρ w ( z ) ) 2 d ρ } { z + n = 1 n ! ( β 1 ) n ( β m ) n ( α 1 ) n ( α l ) n z n + 1 } ( z U ) ,
(2.12)

where w ( z ) is analytic in U with w ( 0 ) = 0 and | w ( z ) | < 1 ( z U ).

Proof From (2.9) and (2.2), we have
H m l ( α 1 ) f ( z ) = z exp { 2 cos α e i α π 2 ( 1 + μ cos α ) 0 1 1 ρ ( log 1 + ρ w ( z ) 1 ρ w ( z ) ) 2 d ρ } ( z U ) .
(2.13)
For
α j C { 0 , 1 , 2 , } ( j = 1 , 2 , , l ) ,

from (2.13) and (1.7), we obtain (2.12). □

3 Properties of the class S m l ( α 1 , α , μ )

Theorem 2 Let f ( z ) S m l ( α 1 , α , μ ) . Then
H m l ( α 1 ) f ( z ) S ( 1 2 ( 1 μ cos α ) )
(3.1)

and the order 1 2 ( 1 μ cos α ) is sharp.

Proof Let h ( z ) be given by (2.2). It follows from the proof of Theorem 1 that
h ( U ) = { w = u + i v : u = cos α 2 ( cos α + μ ) v 2 + 1 2 ( 1 μ cos α ) } .
(3.2)
By using (3.2), we find that
min | z | = 1 ( z 1 ) { e i α ( h ( z ) cos α + i sin α ) } = min v ( , + ) g ( v ) cos α + sin 2 α ,
where
g ( v ) = cos 2 α 2 ( cos α + μ ) v 2 + cos α μ 2 + v sin α ( < v < + ) .
Since
g ( v ) = cos 2 α cos α + μ v + sin α , g ( v ) > 0 ,
the function g ( v ) attains its minimum value at
v 0 = ( cos α + μ ) sin α cos 2 α .
Thus
min | z | = 1 ( z 1 ) { e i α ( h ( z ) cos α + i sin α ) } = g ( v 0 ) cos α + sin 2 α = sin 2 α ( cos α + μ ) 2 cos α + cos α ( cos α μ ) 2 + sin 2 α = 1 2 ( 1 μ cos α ) .
(3.3)
If f ( z ) S m l ( α 1 , α , μ ) , then we deduce from Theorem 1 and (3.3) that
z ( H m l ( α 1 ) f ( z ) ) H m l ( α 1 ) f ( z ) > 1 2 ( 1 μ cos α ) ( z U )

and the order 1 2 ( 1 μ cos α ) in (3.1) is sharp for the function f 0 ( z ) defined by (2.11). □

Theorem 3 Let f ( z ) S m l ( α 1 , α , μ ) and 1 2 ( 1 μ cos α ) β < 1 . Then H m l ( α 1 ) f ( z ) is α-spirallike of order β in | z | < ρ , where
ρ = ρ ( β , α , μ ) = ( tan ( π 4 2 cos α ( 1 β ) cos α + μ ) ) 2 .
(3.4)

The result is sharp.

Proof From (3.4) and (2.2) we have
0 < ρ 1 ( 1 2 ( 1 μ cos α ) β < 1 )
and
h ( ρ ) = 1 + 2 π 2 ( 1 + μ cos α ) ( log 1 + i ρ 1 i ρ ) 2 = 1 8 π 2 ( 1 + μ cos α ) ( arctan ρ ) 2 = β .
Hence
inf | z | < ρ h ( z ) = h ( ρ ) = β .
(3.5)
Let f ( z ) S m l ( α 1 , α , μ ) . Then it follows from Theorem 1 and (3.5) that
{ e i α z ( H m l ( α 1 ) f ( z ) ) H m l ( α 1 ) f ( z ) } > β cos α ( | z | < ρ ) ,

that is, H m l ( α 1 ) f ( z ) is α-spirallike of order β in | z | < ρ . Also, the result is sharp for the function f 0 ( z ) defined by (2.11). □

Setting β = 1 2 ( 1 μ cos α ) , Theorem 3 reduces to the following.

Corollary 3 Let f ( z ) S m l ( α 1 , α , μ ) . Then H m l ( α 1 ) f ( z ) is α-spirallike of order 1 2 ( 1 μ cos α ) in  U . The result is sharp.

For β 1 , a function f ( z ) A is said to be prestarlike of order β in U if
{ z ( 1 z ) 2 ( 1 β ) f ( z ) S ( β ) , β < 1 , f ( z ) z > 1 2 , β = 1 ,
(3.6)

(see [20]). We denote this class by R ( β ) ( β 1 ). The following lemma is due to Ruscheweyh [[20], p.54].

Lemma 1 Let β 1 , f ( z ) R ( β ) and g ( z ) S ( β ) . Then, for any analytic function F ( z ) in  U ,
f ( F g ) f g ( U ) co ¯ ( F ( U ) ) ,

where co ¯ ( F ( U ) ) denotes the convex hull of F ( U ) .

Applying the lemma, we derive Theorems 4 and 5 below.

Theorem 4 Let
α 1 > 0 and α 1 max { α 1 , 1 + μ cos α } .
(3.7)
Then
S m l ( α 1 , α , μ ) S m l ( α 1 , α , μ ) .
(3.8)

Proof

Define
ϕ ( z ) = z + n = 1 ( α 1 ) n ( α 1 ) n z n + 1 ( z U )
for α 1 and α 1 satisfying (3.7). Then ϕ ( z ) A and
z ( 1 z ) α 1 ϕ ( z ) = z ( 1 z ) α 1 ( z U ) .
(3.9)
In view of α 1 α 1 > 0 , it follows from (3.9) that
z ( 1 z ) α 1 ϕ ( z ) S ( 1 α 1 2 ) S ( 1 α 1 2 ) ,
which implies that
ϕ ( z ) R ( 1 α 1 2 ) .
(3.10)
Also, for f ( z ) A , (3.9) leads to
{ H m l ( α 1 ) f ( z ) = ϕ ( z ) H m l ( α 1 ) f ( z ) , z ( H m l ( α 1 ) f ( z ) ) = ϕ ( z ) ( z ( H m l ( α 1 ) f ( z ) ) ) .
(3.11)
Let f ( z ) S m l ( α 1 , α , μ ) . Then, by Theorems 1 and 2, we have
{ F ( z ) = z ( H m l ( α 1 ) f ( z ) ) H m l ( α 1 ) f ( z ) e i α ( h ( z ) cos α + i sin α ) , H m l ( α 1 ) f ( z ) S ( 1 2 ( 1 μ cos α ) ) S ( 1 α 1 2 )
(3.12)
for h ( z ) given by (2.2) and α 1 1 + μ cos α . Since the function e i α ( h ( z ) cos α + i sin α ) is convex univalent in U , from (3.10), (3.11), (3.12) and the lemma, we deduce that
z ( H m l ( α 1 ) f ( z ) ) H m l ( α 1 ) f ( z ) = ϕ ( z ) ( z ( H m l ( α 1 ) f ( z ) ) ) ϕ ( z ) H m l ( α 1 ) f ( z ) = ϕ ( z ) ( F ( z ) H m l ( α 1 ) f ( z ) ϕ ( z ) H m l ( α 1 ) f ( z ) e i α ( h ( z ) cos α + i sin α ) .

Therefore, by Theorem 1, f ( z ) S m l ( α 1 , α , μ ) and (3.8) is proved. □

Theorem 5 Let f ( z ) S m l ( α 1 , α , μ ) and g ( z ) R ( 1 2 ( 1 μ cos α ) ) . Then
( f g ) ( z ) S m l ( α 1 , α , μ ) .
(3.13)
Proof Let f ( z ) S m l ( α 1 , α , μ ) . According to Theorems 1 and 2, we have
F ( z ) = z ( H m l ( α 1 ) f ( z ) ) H m l ( α 1 ) f ( z ) e i α ( h ( z ) cos α + i sin α )
and
H m l ( α 1 ) f ( z ) S ( 1 2 ( 1 μ cos α ) ) .
(3.14)
If we put ϕ ( z ) = ( f g ) ( z ) , then
z ( H m l ( α 1 ) ϕ ( z ) ) H m l ( α 1 ) ϕ ( z ) = g ( z ) ( z ( H m l ( α 1 ) f ( z ) ) ) g ( z ) H m l ( α 1 ) f ( z ) = g ( z ) ( F ( z ) H m l ( α 1 ) f ( z ) ) g ( z ) H m l ( α 1 ) f ( z ) ( z U )
(3.15)

for g ( z ) R ( 1 2 ( 1 μ cos α ) ) .

In view of (3.14) and (3.15), an application of the lemma leads to
z ( H m l ( α 1 ) ϕ ( z ) ) H m l ( α 1 ) ϕ ( z ) e i α ( h ( z ) cos α + i sin α ) .

Consequently, by applying Theorem 1, ϕ ( z ) S m l ( α 1 , α , μ ) and the proof of (3.13) is completed. □

Note that R ( 1 2 ) = S ( 1 2 ) . Since R ( β 1 ) R ( β 2 ) for β 1 β 2 1 (see [[15], p.49], we have
K = R ( 0 ) R ( 1 2 ( 1 μ cos α ) ) ( cos α < μ cos α ) .

Thus Theorem 5 yields the following.

Corollary 4
  1. (i)
    If f ( z ) S m l ( α 1 , α , 0 ) and g ( z ) S ( 1 2 ) , then
    ( f g ) ( z ) S m l ( α 1 , α , 0 ) .
     
  2. (ii)
    If f ( z ) S m l ( α 1 , α , μ ) with cos α < μ cos α and g ( z ) K , then
    ( f g ) ( z ) S m l ( α 1 , α , μ ) .
     
Theorem 6 The function f ( z ) A defined by
H m l ( α 1 ) f ( z ) = z ( 1 b z ) 2 cos α e i α ( z U )
(3.16)
belongs to the class S m l ( α 1 , α , μ ) , where
α j C { 0 , 1 , 2 , } ( j = 1 , 2 , , l ) ,
b is complex and
| b | { cos α + μ 3 cos α μ ( cos α < μ < cos α 3 ) , μ cos α + μ ( μ cos α 3 ) .
(3.17)

The result is sharp, that is, | b | cannot be increased.

Proof For f ( z ) A defined by (3.16) and
α j C { 0 , 1 , 2 , } ( j = 1 , 2 , , l ) ,
we easily have
e i α z ( H m l ( α 1 ) f ( z ) ) H m l ( α 1 ) f ( z ) = 1 + b z 1 b z cos α + i sin α ( z U ) .
(3.18)
Hence, by Theorem 1, f ( z ) S m l ( α 1 , α , μ ) if and only if
1 + b z 1 b z h ( z ) ,
(3.19)
where h ( z ) is given by (2.2). Clearly, (3.19) is equivalent to
{ w : | w 1 + | b | 2 1 | b | 2 | < 2 | b | 1 | b | 2 } h ( U )
(3.20)
for 0 < | b | < 1 . Let
δ = min { | w 1 + | b | 2 1 | b | 2 | : w h ( U ) } ,
(3.21)
where h ( U ) is given by (3.2). Then we have
{ δ = min { g ( u ) : u 1 2 ( 1 μ cos α ) } , g ( u ) = ( u 1 + | b | 2 1 | b | 2 ) 2 + 2 ( 1 + μ cos α ) ( u cos α μ 2 cos α ) ( u cos α μ 2 cos α ) .
(3.22)
Note that
1 2 ( 1 μ cos α ) < 1 + | b | 2 1 | b | 2 , g ( u ) = 2 ( u ( 2 | b | 2 1 | b | 2 μ cos α ) ) .
(3.23)
(i)If
cos α < μ < cos α 3 and | b | = cos α + μ 3 cos α μ ,
(3.24)
then
1 | b | 1 + | b | = 1 2 ( 1 μ cos α ) , | b | 2 = ( cos α + μ 3 cos α μ ) 2 < cos α + μ 5 cos α + μ ,
and so
2 | b | 2 1 | b | 2 μ cos α < 1 2 ( 1 μ cos α ) .
(3.25)
From (3.22), (3.23) and (3.25), we have g ( u ) > 0 ( u 1 2 ( 1 μ cos α ) ) , and hence
δ = g ( 1 2 ( 1 μ cos α ) ) = 1 + | b | 2 1 | b | 2 1 2 ( 1 μ cos α ) = 2 | b | 1 | b | 2 .
(3.26)
(ii) If
cos α < μ < cos α 3 and cos α + μ 3 cos α μ < | b | < cos α + μ 5 cos α + μ ,
(3.27)
then
1 | b | 1 + | b | < 1 2 ( 1 μ cos α ) and g ( u ) > 0 ( u 1 2 ( 1 μ cos α ) ) .
Hence
δ = g ( 1 2 ( 1 μ cos α ) ) < 2 | b | 1 | b | 2 .
(3.28)
(iii) If
μ cos α 3 and | b | = μ cos α + μ ,
(3.29)
then
| b | 2 = μ cos α + μ cos α + μ 5 cos α + μ ,
and so
2 | b | 2 1 | b | 2 μ cos α 1 2 ( 1 μ cos α ) .
Thus g ( u ) attains its minimum value at
u 0 = 2 | b | 2 1 | b | 2 μ cos α
and
δ = g ( u 0 ) = 2 | b | cos α + μ cos α ( 1 | b | 2 ) = 2 | b | 1 | b | 2 .
(3.30)
(iv) If
μ cos α 3 and μ cos α + μ < | b | < 1 ,
(3.31)
then from (iii) we easily have
δ = g ( u 0 ) < 2 | b | 1 | b | 2 .
(3.32)

Now, by virtue of (3.19), (3.20), (3.21), and (i)-(iv), we have proved the theorem. □

Theorem 7 Let
f ( z ) = z + n = 2 a n z n S m l ( α 1 , α , μ ) ,
where
α j C { 0 , 1 , 2 , 3 , } ( j = 1 , 2 , , l ) .
Then
| a 2 | 8 ( cos α + μ ) π 2 | β 1 β m α 1 α l | .
(3.33)

The result is sharp.

Proof It can be easily verified that, for z U ,
z ( H m l ( α 1 ) f ( z ) ) H m l ( α 1 ) f ( z ) = 1 + α 1 α l β 1 β m a 2 z +
(3.34)
and
h ( z ) = 1 + 8 z π 2 ( 1 + μ cos α ) ( n = 1 z n 1 2 n 1 ) 2 = 1 + 8 π 2 ( 1 + μ cos α ) n = 1 ( 1 n ν = 0 n 1 1 2 ν + 1 ) z n = 1 + 8 π 2 ( 1 + μ cos α ) z + ,
(3.35)
where
f ( z ) = z + a 2 z 2 + S m l ( α 1 , α , μ )
and h ( z ) is given by (2.2). From (3.34), (3.35) and Theorem 1, we obtain
π 2 e i α 8 ( cos α + μ ) ( z ( H m l ( α 1 ) f ( z ) ) H m l ( α 1 ) f ( z ) 1 ) = π 2 e i α α 1 α l 8 ( cos α + μ ) β 1 β m a 2 z + π 2 cos α 8 ( cos α + μ ) ( h ( z ) 1 ) K .
(3.36)
It is the well-known Rogosinski result (cf. [[22], p.195]) that if
g ( z ) = n = 1 b n z n

is analytic in U , g ( z ) ϕ ( z ) and ϕ ( z ) K , then | b n | 1 ( n N ). Hence (3.33) follows from (3.36) at once. □

The estimate (3.33) is sharp since equality is attained for the function f 0 ( z ) defined by (2.11).

Declarations

Acknowledgements

Dedicated to Professor Hari M Srivastava.

This work was partially supported by the National Natural Science Foundation of China (Grant No. 11171045).

Authors’ Affiliations

(1)
Department of Mathematics, Changshu Institute of Technology
(2)
Department of Mathematics, Suzhou University
(3)
Department of Mathematics, Rzeszów University of Technology

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