Open Access

Properties for certain subclasses of analytic functions with nonzero coefficients

Journal of Inequalities and Applications20142014:506

https://doi.org/10.1186/1029-242X-2014-506

Received: 18 July 2014

Accepted: 2 December 2014

Published: 12 December 2014

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

Let A denote the class of functions of the form
f ( z ) = z + n = 2 a n z n ( a n 0 ) ,
(1.1)

which are analytic in the open unit disk U = { z C : | z | < 1 } . We also denote by S the class of all functions in A which are univalent in U .

A function f A is said to be in the class t ( A , B , α ) if
| f ( z ) 1 t ( A B ) B ( f ( z ) 1 ) | < α ( z U ) ,
(1.2)

where A and B are complex numbers with A B , t C { 0 } , and α is a positive real number.

In particular, for some real numbers A and B with 1 B < A 1 and α = 1 without any restriction of the coefficients a n ( n N = { 1 , 2 , } ) the class t ( A , B , α ) was introduced by Dixit and Pal [1]. Moreover, by giving specific values t, A, B, and α in (1.2), we obtain subclasses studied by various researchers in earlier works (see [26]).

A function f A is said to be in the class UST ( α ) if
{ f ( z ) f ( ξ ) ( z ξ ) f ( z ) } > α ( z × ξ U × U ; 0 α < 1 ) .
Furthermore, a function f A is said to be in the class UCV ( α ) if
{ 1 + ( z ξ ) f ( z ) f ( z ) } > α ( z × ξ U × U ; 0 α < 1 ) .

The classes UST ( 0 ) UST and UCV ( 0 ) UCV are introduced by Goodman [7, 8] (they are called the classes of uniformly starlike and uniformly convex functions, respectively). The classes of uniformly starlike and uniformly convex functions have been extensively studied by Ma and Minda [9] and Rønning [10].

Let us consider the Gaussian hypergeometric function F ( a , b ; c ; z ) defined by
F ( a , b ; c ; z ) = n = 0 ( a ) n ( b ) n ( c ) n z n ( a , b , c C ; c 0 , 1 , 2 , ; z U ) ,
where ( ν ) n is the Pochhammer symbol (or the shifted factorial) defined (in terms of the gamma function) by
( ν ) n : = Γ ( ν + n ) Γ ( ν ) = { 1 if  n = 0  and  ν C { 0 } , ν ( ν + 1 ) ( ν + n 1 ) if  n N  and  ν C .
We note that F ( a , b ; c ; z ) = F ( b , a ; c ; z ) and
F ( a , b ; c ; 1 ) = Γ ( c a b ) Γ ( c ) Γ ( c a ) Γ ( c b ) ( { c a b } > 0 ) .

We also recall (see [11, 12]) that the function F ( a , b ; c ; z ) is bounded if { c a b } > 0 , and it has a pole at z = 1 if { c a b } 0 . Moreover, univalence, starlikeness and convexity properties of z F ( a , b ; c ; z ) have been extensively studied by Ponnusamy and Vuorinen [13] and Ruscheweyh and Singh [14].

For f A , we define the operator I a , b ; c f by
I a , b ; c f ( z ) = z F ( a , b ; c ; z ) f ( z ) ,
(1.3)

where denotes the usual Hadamard product (or convolution) of power series. For a special case of the operator I a , b ; c f , we can refer to the paper by Swaminathan [15] 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 classes t ( A , B , α ) , UST ( α ) , and UCV ( α ) , respectively. Moreover, we find a condition for univalency of the operator I a , b ; c f defined by (1.3).

2 Main result

Theorem 1 Let a function f of the form (1.1) be in the class t ( A , B , α ) with a n = | a n | e i ( 3 n + 1 ) π 2 ( n N { 1 } ), then
n = 2 n ( 1 α | B | ) | a n | α | t | | A B | .
(2.1)
Proof From the definition of t ( A , B , α ) , we have
| f ( z ) 1 | < α | t ( A B ) B ( f ( z ) 1 ) | ( z U ) ,
and so
| n = 2 n a n z n 1 | < α | t ( A B ) B n = 2 n a n z n 1 | .
(2.2)
If we take z = r e i π 2 , then we see that
a n z n 1 = | a n | r n 1 ( 0 r < 1 ) .
(2.3)
Then, by using (2.3) to (2.2), we have
n = 2 n | a n | r n 1 < α | t ( A B ) B n = 2 n | a n | r n 1 | < α | t ( A B ) | + α | B | n = 2 n | a n | r n 1 ,
or, equivalently,
n = 2 n ( 1 α | B | ) | a n | r n 1 < α | t | | A B | .
(2.4)

Letting r 1 in (2.4), we have the inequality (2.1). Thus we complete the proof of Theorem 1. □

Theorem 2 Let a function f of the form (1.1) be in the class A . If
n = 2 n ( 1 + α | B | ) | a n | α | t | | A B | ,
(2.5)
where A and B are complex numbers with A B , t C { 0 } , and α is a positive real number, then f t ( A , B , α ) . The result is sharp for the function defined by
f ( z ) = z + n = 2 α t ( A B ) ϵ n 2 ( n 1 ) ( 1 + α | B | ) z n ( A , B C ; A B ; t C { 0 } ; | ϵ | = 1 ; z U ) .
Proof In view of the definition of t ( A , B , α ) , it suffices to prove that
| f ( z ) 1 | < α | t ( A B ) B ( f ( z ) 1 ) | ( z U ) .
(2.6)
From (2.6), we have
| n = 2 n a n z n 1 | < α | t ( A B ) B n = 2 n a n z n 1 | .
Hence it is sufficient to show that
n = 2 n | a n | r n 1 < α ( | t | | A B | | B | n = 2 n | a n | r n 1 ) ,
which is equivalent to the relation
n = 2 n ( 1 + α | B | ) | a n | r n 1 < α | t | | A B | .
(2.7)
Letting r 1 in (2.7), we have
n = 2 n ( 1 + α | B | ) | a n | α | t | | A B | .

Therefore, by the assumption (2.5), we prove that f t ( A , B , α ) . □

Theorem 3 Let a function f of the form (1.1) be in the class A . If
n = 2 ( ( 3 α ) n 2 ) | a n | 1 α ( 0 α < 1 ) ,
(2.8)
then f UST ( α ) . The result is sharp for the function
f ( z ) = z + n = 2 ( 1 α ) ϵ n ( n 1 ) ( ( 3 α ) n 2 ) z n ( 0 α < 1 ; | ϵ | = 1 ) .
Proof It suffices to show that
| f ( z ) f ( ξ ) ( z ξ ) f ( z ) 1 | 1 α ( 0 α < 1 ; ( z , ξ ) U × U ) .
We have
| f ( z ) f ( ξ ) ( z ξ ) f ( z ) 1 | = | n = 2 a n ( ξ n 1 + z ξ n 2 + + z n 1 ) n = 2 n a n z n 1 1 + n = 2 n a n z n 1 | n = 2 2 ( n 1 ) | a n | 1 n = 2 n | a n | ,

which is bounded by 1 α if the assumption (2.8) is satisfied. Thus we complete the proof of Theorem 3. □

Theorem 4 Let a function f of the form (1.1) be in the class A . If
n = 2 n ( 2 n 1 α ) | a n | 1 α ( 0 α < 1 ) ,
(2.9)
then f UCV ( α ) . The result is sharp for the function
f ( z ) = z + n = 2 ( 1 α ) ϵ n 2 ( n 1 ) ( 2 n 1 α ) z n ( 0 α < 1 ; | ϵ | = 1 ; z U ) .
Proof It suffices to show that
| ( z φ ) f ( z ) f ( z ) | < 1 α ( 0 α < 1 ; ( z , φ ) U × U ) .
We have
| ( z φ ) f ( z ) f ( z ) | = | ( z φ ) n = 2 n ( n 1 ) a n z n 2 1 n = 2 n a n z n 1 | 2 n = 1 n ( n 1 ) | a n | 1 n = 2 n | a n | ,

which is bounded by 1 α if the assumption (2.9) is satisfied. Thus we complete the proof of Theorem 4. □

Theorem 5 Let a , b C { 0 } and c > | a | + | b | . If f t ( A , B , α ) with a n = | a n | e i ( 3 n + 1 ) π 2 , 0 < | B | < 1 , and
Γ ( c | a | | b | ) Γ ( c ) Γ ( c | a | ) Γ ( c | b | ) 1 α | B | 1 + α | B | + 1 ,

then I a , b ; c f t ( A , B , α ) , where the operator I a , b ; c f is defined by (1.3).

Proof We want to prove from Theorem 2 that
T 1 : = n = 2 n ( 1 + α | B | ) | A n | α | t | | A B | ,
where
A n = ( a ) n 1 ( b ) n 1 ( c ) n 1 ( 1 ) n 1 a n .
Since, from Theorem 1,
| a n | α | t | | A B | n ( 1 α | B | ) , T 1 α | t | | A B | ( 1 + α | B | ) 1 α | B | n = 2 ( | a | ) n 1 ( | b | ) n 1 ( c ) n 1 ( 1 ) n 1 T 1 = α | t | | A B | ( 1 + α | B | ) 1 α | B | ( n = 0 ( | a | ) n ( | b | ) n ( c ) n ( 1 ) n 1 ) T 1 = α | t | | A B | ( 1 + α | B | ) 1 α | B | ( Γ ( c | a | | b | ) Γ ( c ) Γ ( c | a | ) Γ ( c | b | ) 1 ) T 1 α | t | | A B | ,

which completes the proof of Theorem 5. □

We introduce the following lemma which is needed for the proof of the next theorem.

Lemma 1 [16]

Let ω be regular in the unit disk U with ω ( 0 ) = 0 . Then, if | ω ( z ) | attains a maximum value on the circle | z | = r ( 0 r < 1 ) at a point z 0 , we can write
z 0 ω ( z 0 ) = k ω ( z 0 ) ( k 1 ) .
Theorem 6 Let a function f of the form (1.1) be in the class A . Assume
| ( I a , b ; c f ( z ) ) 1 1 α | β | z ( I a , b ; c f ( z ) ) ( I a , b ; c f ( z ) ) α | γ < 1 2 γ ( z U )
(2.10)
for some real α ( 0 α < 1 ), β > 0 , and γ > 0 . Then
| ( I a , b ; c f ( z ) ) 1 | < 1 α ( z U ) .
(2.11)
Proof Let us define ω by
ω ( z ) = ( I a , b ; c f ( z ) ) 1 1 α ( z U ) .
Then it follows that ω is analytic in U with ω ( 0 ) = 0 . By (2.10), we have
| ω ( z ) | β | z ω ( z ) ω ( z ) + 1 | γ = | ω ( z ) | β + γ | z ω ( z ) ω ( z ) 1 ω ( z ) + 1 | γ < 1 2 γ ( z U ) .
(2.12)
Suppose that there exists a point z 0 U such that
max | z | | z 0 | | ω ( z ) | = | ω ( z 0 ) | = 1 .
Then, by Lemma 1, we can put
z 0 ω ( z 0 ) ω ( z 0 ) = k 1 .
Hence, we obtain
| ω ( z 0 ) | β | z 0 ω ( z 0 ) ω ( z 0 ) + 1 | γ = | z 0 ω ( z 0 ) ω ( z 0 ) + 1 | γ ( k 2 ) γ 1 2 γ ,
which contradicts the condition (2.12). This shows that
| ω ( z ) | = | ( I a , b ; c f ( z ) ) 1 1 α | < 1 ( z U ) .

Thus we complete the proof of Theorem 6. □

Remark 1 The condition (2.11) in Theorem 6 implies that
{ ( I a , b ; c f ( z ) ) } > 0 ( z U ) .

Therefore, by the Noshiro-Warschawski theorem [17], the operator I a , b ; c f is univalent in U under the restrictions of Theorem 6.

Declarations

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).

Authors’ Affiliations

(1)
Department of Applied Mathematics, Pukyong National University
(2)
Department of Mathematics, Faculty of Education, Yamato University

References

  1. Dixit KK, Pal SK: On a class of univalent functions related to complex order. Indian J. Pure Appl. Math. 1995,26(9):889-896.MathSciNetGoogle Scholar
  2. Caplinger TR, Causey WM: A class of univalent functions. Proc. Am. Math. Soc. 1973, 39: 357-361. 10.1090/S0002-9939-1973-0320294-4MathSciNetView ArticleGoogle Scholar
  3. Kim JA, Cho NE: Properties of convolutions for hypergeometric series with univalent functions. Adv. Differ. Equ. 2013. Article ID 101, 2013: Article ID 101Google Scholar
  4. Dashrath: On some classes related to spiral-like univalent and multivalent functions. PhD thesis, Kanpur University, Kanpur (1984)Google Scholar
  5. Padmanabhan, KS: On a certain class of functions whose derivatives have a positive real part in the unit disc. Ann. Pol. Math. 23, 73-81 (1970/1971)Google Scholar
  6. Ponnusamy S, Rønning F: Starlikeness properties for convolutions involving hypergeometric series. Ann. Univ. Mariae Curie-Skłodowska, Sect. A 1998, 52: 141-155.Google Scholar
  7. Goodman AW: On uniformly convex functions. Ann. Pol. Math. 1991, 56: 87-92.Google Scholar
  8. Goodman AW: On uniformly starlike functions. J. Math. Anal. Appl. 1991, 155: 364-370. 10.1016/0022-247X(91)90006-LMathSciNetView ArticleGoogle Scholar
  9. Ma W, Minda D: Uniformly convex functions. Ann. Pol. Math. 1992, 57: 166-175.MathSciNetGoogle Scholar
  10. Rønning F: Uniformly convex functions and a corresponding class of starlike functions. Proc. Am. Math. Soc. 1993, 118: 190-196.View ArticleGoogle Scholar
  11. Owa S, Srivastava HM: Univalent and starlike generalized hypergeometric functions. Can. J. Math. 1987, 39: 1057-1077. 10.4153/CJM-1987-054-3MathSciNetView ArticleGoogle Scholar
  12. Whitteaker ET, Watson GN: A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and Analytic Functions: With an Account of the Principal Transcendental Functions. 4th edition. Cambridge University Press, Cambridge; 1927.Google Scholar
  13. Ponnusamy, S, Vuorinen, M: Univalence and convexity properties for Gaussian hypergeometric functions. Preprint 82, Department of Mathematics, University of Helsinki (1995)Google Scholar
  14. Ruscheweyh S, Singh V: On the order of starlikeness of hypergeometric functions. J. Math. Anal. Appl. 1986, 113: 1-11. 10.1016/0022-247X(86)90329-XMathSciNetView ArticleGoogle Scholar
  15. Swaminathan A: Certain sufficiency conditions on Gaussian hypergeometric functions. J. Inequal. Pure Appl. Math. 2004. Article ID 83, 5: Article ID 83Google Scholar
  16. Jack IS: Functions starlike and convex of order α . J. Lond. Math. Soc. 1971, 3: 469-474.MathSciNetView ArticleGoogle Scholar
  17. Duren PL: Univalent Functions. Springer, New York; 1983.Google Scholar

Copyright

© Lee et al.; licensee Springer. 2014

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