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Goswami et al. Journal of Inequalities and Applications 2014, 2014:431
http://www.journalofinequalitiesandapplications.com/content/2014/1/431
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Open Access
Extensions of some generalized conditions
for starlikeness and convexity
Pranay Goswami1* , Teodor Bulboac˘a2 and Badr S Alkahtani3
*
Correspondence:
[email protected]
1
Department of Mathematics,
School of Liberal Studies, Ambedkar
University Delhi, Delhi, 110007, India
Full list of author information is
available at the end of the article
Abstract
In the present paper we obtain generalized sufficient conditions for the starlikeness
and convexity of some normalized analytic functions; the results extend those
recently obtained in the work of Uyanik and Owa (J. Inequal. Appl. 2011:87, 2011).
MSC: 30C45
Keywords: analytic functions; starlike and convex functions; n-starlike and n-convex
functions; coefficient estimates
1 Introduction and preliminaries
Let H(U) be the class of holomorphic functions in the unit disk U = {z ∈ C : |z| < }, and
define
An = f ∈ H(U) : f (z) = z + an+ zn+ + · · · , z ∈ U ,
n ∈ N.
Also, let
zf (z)
> , z ∈ U
Sn∗ = f ∈ An : Re
f (z)
be the class of n-starlike functions in U, and
zf (z)
> , z ∈ U
Cn = f ∈ An : Re  + f (z)
be the class of n-convex functions in U. The notations S ∗ := S∗ and C := C are well known,
where S ∗ and C represent, respectively, the class of normalized starlike and normalized
convex functions in the unit disc U.
In the recent years, many authors have obtained some sufficient conditions for starlikeness and convexity, that can be found, for example, in [–] etc., together with many
references. In the present paper, by using some simple integral inequalities, we generalize
the results of [] and we give some interesting special cases of our results. The methods
used here are frequently used in complex analysis and operator theory on spaces of analytic functions; among others, for example, in obtaining estimations for point evaluation
operators on spaces of analytic functions of one or several variables containing derivatives
in the definition of the spaces (see, for example, [] and []).
©2014 Goswami et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction
in any medium, provided the original work is properly cited.
Goswami et al. Journal of Inequalities and Applications 2014, 2014:431
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Page 2 of 13
To obtain the generalized conditions for n-starlikeness and n-convexity, presented in
the following sections, we shall require the following lemma.
Lemma . [, ] If f ∈ An and
f (z) –  < n + 
,
(n + ) + 
z ∈ U,
(.)
then f ∈ Sn∗ .
2 Generalized conditions for n-starlikeness
Using the previous lemma, we will prove the next sufficient condition for n-starlikeness.
Theorem . Let f ∈ An and suppose that
(j ) f  (z) ≤ n + 
– Mn (j ),
(n + ) + 
z ∈ U,
(.)
for some integer j ≥ , with
,
Mn (j ) := j –n–
k=
|f
(n+k)
if  ≤ j ≤ n + ,
()|, if j ≥ n + .
(.)
Then f ∈ Sn∗ .
Proof Since f ∈ An , we have
f (k) () =  for  ≤ k ≤ n.
(.)
(i) First, we will prove that (.) implies f ∈ Sn∗ for j = . Thus, if
f (z) ≤ n + 
,
(n + ) + 
z ∈ U,
then we have
z
f (z) –  = f (ζ ) dζ ≤

≤
|z| f ρeiθ dρ

n+
(n + )
+
|z| < n+
(n + ) + 
,
z ∈ U,
and according to Lemma . we deduce that f ∈ Sn∗ .
(ii) Suppose that there exists an integer  ≤ j ≤ n +  such that inequality (.) holds.
Using relation (.), we may easily see that
z
(j –) z (j )
f  (z) = f  (ζ ) dζ + f (j –) () = f (j ) (ζ ) dζ 
≤


|z| n+
f (j ) ρeiθ dρ ≤ n + 
|z| < ,

(n + ) + 
(n + ) + 
z ∈ U.
Goswami et al. Journal of Inequalities and Applications 2014, 2014:431
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Consequently, if inequality (.) holds for some integer  ≤ j ≤ n + , then
(j) f (z) < n + 
,
(n + ) + 
z ∈ U, for all  ≤ j ≤ j – .
Using the fact that we have already proved that if (.) holds for j = , then f ∈ Sn∗ , from
the above remark it follows that our result holds for any integer  ≤ j ≤ n + .
(iii) Now we will prove that the result holds for j = n + . Thus, a simple computation
shows that
(n+) z (n+)
f
(z) = f
(ζ ) dζ + f (n+) ()

z
|z|
(n+) iθ f
≤ f (n+) (ζ ) dζ + f (n+) () ≤
ρe dρ + f (n+) ()


n+
n+
– Mn (n + ) |z| + f (n+) () < , z ∈ U,
≤ 
(n + ) + 
(n + ) + 
that is,
(n+) n+
f
,
(z) < (n + ) + 
z ∈ U,
and using the result of the part (ii) of our proof for j = n + , it follows that f ∈ Sn∗ .
(iv) Finally, supposing that there exists an integer j ≥ n +  such that (.) holds, it
follows that
(j –) z (j )
f  (z) = f  (ζ ) dζ + f (j –) ()

z
|z|
(j ) iθ f  ρe dρ + f (j –) ()
≤ f (j ) (ζ ) dζ + f (j –) () ≤


n+
– Mn (j ) |z| + f (j –) ()
≤ 
(n + ) + 
<
n+
(n + ) + 
– Mn (j – ),
z ∈ U.
Consequently, if inequality (.) holds for some integer j ≥ n + , then
(j) f (z) < n + 
– Mn (j),
(n + ) + 
z ∈ U, for all n +  ≤ j ≤ j – .
Since we have already proved that if (.) holds for j = n +  then f ∈ Sn∗ , from the above
remark the second part of our result follows, and the theorem is completely proved. Remark . If we put n =  in the above result, we get [, Theorem .].
3 Generalized conditions for n-convexity
In order to prove our main result, first we give the following two lemmas.
Page 3 of 13
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Page 4 of 13
Lemma . Let f ∈ An and suppose that
(j ) f  (z) ≤ n + 
,
 (n + ) + 
z ∈ U,
(.)
for some integer  ≤ j ≤ n + . Then f ∈ Cn .
Proof (i) First, we will show that this implication holds for j = . Letting F(z) = zf (z), then
f ∈ Cn if and only if F ∈ Sn∗ . If
f (z) ≤ n + 
,
 (n + ) + 
z ∈ U,
a simple computation shows that
F (z) –  = zf (z) + f (z) –  ≤ zf (z) + f (z) – 
|z|
z
iθ f ρe dρ
≤ zf (z) + f (ζ ) dζ ≤ zf (z) +


n+
n+
|z| + |z| < ,
≤ 

 (n + ) + 
 (n + ) + 
(n + ) + 
n+
z ∈ U,
and by Lemma . we deduce that F ∈ Sn∗ , i.e., f ∈ Cn .
(ii) Suppose that there exists an integer  ≤ j ≤ n +  such that inequality (.) holds.
Using relation (.), we may easily see that
z
(j –) z (j )
f  (z) = f  (ζ ) dζ + f (j –) () = f (j ) (ζ ) dζ 
≤


|z| n+
f (j ) ρeiθ dρ ≤ n + 
|z| < ,

 (n + ) + 
 (n + ) + 
z ∈ U.
Thus, we have obtained that if inequality (.) holds for some integer  ≤ j ≤ n + , then
(j) f (z) < n + 
,
 (n + ) + 
z ∈ U, for all  ≤ j ≤ j – .
Since we have already proved that if (.) holds for j =  then f ∈ Cn , our result follows
from the above remark.
Lemma . Let f ∈ An and suppose that
(j +) f  (z) ≤
n+

– Nn (j ) ,
j + 
(n + ) + 
z ∈ U,
(.)
for some integer j ≥ n + , with
(j ) if j = n + ,
(n + )|f (n+) ()|,
Nn (j ) := Mn (j ) + j f  () = j –n– (n+k)
(j )
|f
()|
+
j
|f
()|,
if
j ≥ n + ,

k=
where Mn (j ) is given by (.). Then f ∈ Cn .
(.)
Goswami et al. Journal of Inequalities and Applications 2014, 2014:431
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Page 5 of 13
Proof Suppose that there exists an integer j ≥ n +  such that inequality (.) holds. If we
let F(z) = zf (z), then f ∈ Cn if and only if F ∈ Sn∗ . Since
F (j ) (z) = zf (j +) (z) + j f (j ) (z),
we obtain that
(j ) (j +)
F  (z) = zf  (z) + k f (j ) (z)
z
≤ |z|f (j +) (z) + j f (j +) (ζ ) dζ + f (j ) ()
≤ |z|f (j +) (z) + j

f (j +) ρeiθ dρ + j f (j ) ()
|z| 
≤ (j + )|z|f (j +) (z) + j f (j ) ()
<
n+
(n + )
n+
– Nn (j ) + j f (j ) () = – Mn (j ),
+
(n + ) + 
z ∈ U.
Thus, we have obtained that if inequality (.) holds for some integer j ≥ n + , then
(j ) F  (z) < n + 
– Mn (j ),
(n + ) + 
z ∈ U,
and from Theorem . we deduce that F ∈ Sn∗ , hence f ∈ Cn , which completes the proof of
the lemma.
Combining Lemma . and Lemma ., we obtain the next conclusion for the nconvexity sufficient conditions.
Theorem . Let f ∈ An and suppose that
(j +) f  (z) ≤ Pn (j ),
z ∈ U,
for some integer j ≥ , with
Pn (j ) :=
⎧
⎨ √ n+  ,
(n+) +
⎩ j + ( √ n+

(n+) +
if  ≤ j ≤ n,

– Nn (j )),
if j ≥ n + ,
where Nn (j ) is given by (.). Then f ∈ Cn .
Remark . If we put n =  in the above results, we obtain [, Theorem .].
Using a similar method as in the proof of Theorem . and Lemma ., we can easily
derive the following result.
Theorem . Let f ∈ An and suppose that
(j +) f  (z) ≤ Qn (j ) := n + 
– Mn (j + ),
 (n + ) + 
z ∈ U,
(.)
Goswami et al. Journal of Inequalities and Applications 2014, 2014:431
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for some integer j ≥ , where Mn (j + ) is defined by (.), i.e.,
,
Mn (j + ) = j –n
k=
|f
(n+k)
if  ≤ j ≤ n,
()|, if j ≥ n + .
Then f ∈ Cn .
Proof (i) For  ≤ j ≤ n, the result follows from Lemma ..
(ii) We will prove that the result holds for j = n + . A simple calculus shows that
(n+) z (n+)
f
(z) = f
(ζ ) dζ + f (n+) ()

z
|z|
(n+) iθ f
≤ f (n+) (ζ ) dζ + f (n+) () ≤
ρe dρ + f (n+) ()


n+
– Mn (n + ) |z| + f (n+) ()
≤

 (n + ) + 
n+
< ,
 (n + ) + 
z ∈ U,
(n+) n+
f
,
(z) <  (n + ) + 
z ∈ U,
that is,
and using the result of the part (i) of our proof for j = n, it follows that f ∈ Cn .
(iii) Supposing that there exists an integer j ≥ n +  such that (.) holds, it follows that
(j ) z (j +)
f  (z) = f  (ζ ) dζ + f (j ) ()

z
|z|
(j ) (j +) iθ (j +)

f  ρe dρ + f (j ) ()
≤ f
(ζ ) dζ + f () ≤


n+
– Mn (j + ) |z| + f (j ) ()
≤
 (n + ) + 
n+
< – Mn (j ),
 (n + ) + 
z ∈ U.
Thus, if inequality (.) holds for some integer j ≥ n + , then
(j+) n+
f
– Mn (j + ),
(z) <  (n + ) + 
z ∈ U, for all n +  ≤ j ≤ j .
Since we have already proved to the part (ii) that if (.) holds for j = n +  then f ∈ Cn ,
our theorem follows from the above remark.
Remark . If we put n =  in the above results, then we get [, Theorem .].
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Page 7 of 13
It remains to determine which of the above last two theorems gives weaker sufficient
conditions for n-convexity, i.e.,
(j +) f  (z) ≤ Rn (j ) := max Pn (j ); Qn (j ) ,
z ∈ U,
implies f ∈ Cn .
(i) First, let us consider the case j ≥ n + .
A simple computation shows that Rn (j ) = Qn (j ) if and only if
Qn (j ) ≥ Pn (j ) and
Qn (j ) ≥ ,
which is equivalent to Mn (j + ) ≤ min{A; B}, where
(j ) n+
j – 

+ f () ,
A :=
j
 (n + ) + 
n+
.
B :=  (n + ) + 
(.)
Moreover, it is easy to check that
min{A; B} = A
⇔
n+
(j – )f (j ) () ≤ ,
 (n + ) + 
min{A; B} = B
⇔
n+
(j – )f (j ) () ≥ .
 (n + ) + 
and
Reversely, Rn (j ) = Pn (j ) if and only if
Qn (j ) ≤ Pn (j ) and
Pn (j ) ≥ ,
that holds whenever A ≤ Mn (j + ) ≤ C, where
C := n+
(n + ) + 
– (j – )f (j ) ().
Moreover, we have
A≤C
⇔
n+
.
(j – )f (j ) () ≥  (n + ) + 
(ii) Secondly, let us consider the case j = n + .
A simple calculus shows that
n+
– f (n+) ()
Rn (n + ) = Qn (n + ) =  (n + ) + 
if and only if
Qn (n + ) ≥ Pn (n + ) and Qn (n + ) ≥ ,
(.)
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Page 8 of 13
which is equivalent to
(n+) n+
n+
n(n + )
f
; .
= () ≤ min 

 (n + ) +   (n + ) + 
 (n + ) + 
Reversely,
Rn (n + ) = Pn (n + ) =
n+
n+
– f (n+) ()
n+
(n + ) + 
if and only if
Qn (n + ) ≤ Pn (n + ) and Pn (n + ) ≥ ,
that holds whenever

n(n + )
≤ f (n+) () ≤ .
 (n + ) + 
(n + ) + 
In order to have
n(n + )

≤
,

 (n + ) + 
(n + ) + 
we obtain that n = , but this case is contained in the first part of (ii).
Now, from Theorem . and Theorem ., we deduce the next conclusion.
Theorem . Let f ∈ An and suppose that
(j +) f  (z) ≤ Rn (j ),
z ∈ U,
(.)
for some integer j ≥ , with
n+
,
(i) Rn (j ) :=  (n + ) + 
if  ≤ j ≤ n,
n+
– f (n+) (),
(ii) Rn (j ) := 
 (n + ) + 
(n+) n+
,
() ≤ and f
 (n + ) + 
(iii)
(iv)
if j = n + 
n+
– Mn (j + ), if j ≥ n + ,
Rn (j ) :=  (n + ) + 
n+
and Mn (j + ) ≤ B,
(j – )f (j +) () >  (n + ) + 
n+
– Mn (j + ), if j ≥ n + ,
Rn (j ) :=  (n + ) + 
n+
and Mn (j + ) ≤ A,
(j – )f (j +) () ≤  (n + ) + 
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(v)
Page 9 of 13
(j ) n+


– Mn (j + ) – (j – ) f () ,
Rn (j ) :=
j + 
(n + ) + 
n+
and A ≤ Mn (j + ) ≤ C,
if j ≥ n + , (j – )f (j +) () ≤  (n + ) + 
where Mn (j ), A, B and C are given by (.), (.) and (.), respectively. Then f ∈ Cn .
4 Special cases
In this section we give some special cases of our main results.
. Taking the polynomial function
fn,m (z) = z + an+ zn+ + · · · + an+m zn+m ,
z ∈ C,
(.)
where m ∈ N, then fn,m ∈ An , deg fn,m = n + m. It is easy to see that
(j )
fn,m
(z) = j !
m n+k
j
k=
z ∈ C, if  ≤ j ≤ n + ,
an+k zn–j +k ,
n+k
an+k zn–j +k ,
n – j + k
(.)
n–j +k (j )
(z) = j !
fn,m
k=j –n
z ∈ C, if n +  ≤ j ≤ n + m,
(.)
and from Theorem . we deduce the following result.
Corollary . Let fn,m be the polynomial function of the form (.) and suppose that for
some integer  ≤ j ≤ n + m, one of the following inequalities holds:
n+
,
(i) sn(j ) (m) ≤ (n + ) + 
(ii)
sn(j ) (m) ≤
n+
(n + ) + 
if  ≤ j ≤ n + ,
j –n–
–
(n + j)!|an+j |,
if n +  ≤ j ≤ n + m,
j=
where
sn(j ) (m) = j !
m n+k
k=
sn(j ) (m) = j !
j
|an+k |,
if  ≤ j ≤ n + ,
m n+k
|an+k |,
n – j + k
if n +  ≤ j ≤ n + m.
(.)
(.)
k=j –n
Then fn,m ∈ Sn∗ .
Taking j = n +  and j = n + m in Corollary ., we deduce that if one of the following
inequalities holds:
(n + )!
m n+k
k=
m
n+
|an+k | ≤ n+
(n + ) + 
n+
,
(n + k)!|an+k | ≤ (n
+ ) + 
k=
,
(.)
(.)
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then fn,m ∈ Sn∗ . It is easy to check that condition (.) is stronger than (.), hence we get
the following.
Example . If fn,m is the polynomial function of the form (.) and
m
(n + k)!|an+k | ≤ k=
n+
(n + ) + 
,
then fn,m ∈ Sn∗ .
Remark . For n =  and m = , the above example reduces to the [, Example .], that
is, if |a | + |a | + |a | ≤ √ , then f, (z) = z + a z + a z + a z ∈ Sn∗ .
According to Theorem . and using the differentiation formulas (.) and (.), we
obtain the next result.
Corollary . Let fn,m be the polynomial function of the form (.) and suppose that for
some integer  ≤ j ≤ n + m – , one of the following inequalities holds:
n+
,
(i) sn(j +) (m) ≤  (n + ) + 
if  ≤ j ≤ n,
n+
– (n + )!|an+ |,
(ii) sn(j +) (m) ≤  (n + ) + 
if j = n + ,
n+
sn(j +) (m) ≤ –
(n + j)!|an+j |,

 (n + ) +  j=
j –n
(iii)
n+
,
if n +  ≤ j ≤ n + m – , (j + )!(j – )|aj + | >  (n + ) + 
and
j –n
(n + j)!|an+j | ≤ B ,
j=
(iv)
sn(j +) (m) ≤
n+
 (n + ) + 
j –n
–
(n + j)!|an+j |,
j=
n+
,
if n +  ≤ j ≤ n + m – , (j + )!(j – )|aj + | ≤  (n + ) + 
and
j –n
(n + j)!|an+j | ≤ A ,
j=
(v)
sn(j +) (m) ≤
j –n
n+

–
(n + j)!|an+j | – j !(j – )|aj | ,
j +   (n + ) +  j=
n+
,
if n +  ≤ j ≤ n + m – , (j + )!(j – )|aj + | ≤  (n + ) + 
and A ≤
j –n
(n + j)!|an+j | ≤ C ,
j=
Goswami et al. Journal of Inequalities and Applications 2014, 2014:431
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(j +)
where sn
Page 11 of 13
(m) are given by (.) and (.), and
j – 
n+
+ j !|aj | ,
A :=
j
 (n + ) + 
n+
,
B :=  (n + ) + 
C := n+
(n + ) + 
– j !(j – )|aj |.
Then fn,m ∈ Cn .
Taking j = n in Corollary ., we deduce the following special case.
Example . If fn,m is the polynomial function of the form (.) and
(n + )!
m n+k
k=
n+
|an+k | ≤ ,
n+
 (n + ) + 
then fn,m ∈ Cn .
Remark . For n =  and m = , the above example reduces to the following one: if |a | +
|a | + |a | ≤ √  , then f, (z) = z + a z + a z + a z ∈ Cn .
. For the function
z
fn (z) = z + α e –
n
zk
k=
k!
=z+
∞
k=
α
zn+k ,
(n + k)!
z ∈ C,
(.)
where α ∈ C, we have fn ∈ An , and we may easily check that
∞
∞
k– (n+) z

= α
f
≤
|α|
= |α|e,
(z)
n
(k – )! (k – )!
k=
z ∈ U.
k=
If we take j = n +  in Theorem . and j = n in Theorem ., we deduce, respectively,
the following.
Example . If fn is the function defined by (.) and
n+
,
|α| ≤ e (n + ) + 
then fn ∈ Sn∗ .
Example . If fn is the function of the form (.) and
|α| ≤
n+
,
e (n + ) + 
then fn ∈ Cn∗ .
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. Considering the function
gn (z) = z + α sin z –
n–
k=
zk+
(–)k
(k + )!
∞
(–)k α k+
z ,
=z+
(k + )!
z ∈ C,
(.)
k=n
where α ∈ C, then gn ∈ An , and
∞
j (n+) z
n+j
g
(z) = α
(–)
n
(j)! j=
∞

e + e–
= |α|
,
(j)!

j=
≤ |α|
z ∈ U.
Taking j = n +  in Theorem . and j = n in Theorem ., we obtain, respectively,
the following.
Example . If gn is the function defined by (.) and
|α| ≤

n + 
,
–
e+e
(n + ) + 
∗
then gn ∈ Sn
.
Example . If gn is the function of the form (.) and
|α| ≤

n + 
,
–
e+e
(n + ) + 
∗
then gn ∈ Cn
.
. Taking the function
hn+ (z) = z + α cos z –
n
k=
=z+
∞
j=
zk
(–)
(k)!
k
n+j
(–) α n+j
z
,
(n + j)!
z ∈ C,
(.)
where α ∈ C, then hn+ ∈ An+ , and
∞
j– (n+) n+j z
h
(–)
n+ (z) = α
(j – )! j=
≤ |α|
∞
j=
e + e–

= |α|
,
(j – )!

z ∈ U.
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If we choose j = n +  in Theorem . and j = n +  in Theorem ., we obtain,
respectively, the following.
Example . If hn+ is the function defined by (.) and
|α| ≤

(n + )
,
e + e– (n + ) + 
∗
.
then hn+ ∈ Sn+
Example . If hn+ is the function of the form (.) and
|α| ≤
(n + )

,
–
e+e
(n + ) + 
∗
.
then hn+ ∈ Cn+
We emphasize that all the above n-starlikeness and n-convexity criteria are very useful
since the direct proofs are too difficult in each of these examples.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Author details
1
Department of Mathematics, School of Liberal Studies, Ambedkar University Delhi, Delhi, 110007, India. 2 Faculty of
Mathematics and Computer Science, Babe¸s-Bolyai University, Cluj-Napoca, 400084, Romania. 3 Department of
Mathematics, College of Science, King Saud University, P.O. Box 1142, Riyadh, 11989, Saudi Arabia.
Acknowledgements
Authors are thankful to reviewers for careful reading of the paper. The third author BS Alkahtani is grateful to King Saud
University, Deanship of Scientific Research, College of Science Research Center to support this project.
Received: 25 September 2014 Accepted: 16 October 2014 Published: 31 Oct 2014
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10.1186/1029-242X-2014-431
Cite this article as: Goswami et al.: Extensions of some generalized conditions for starlikeness and convexity. Journal
of Inequalities and Applications 2014, 2014:431
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