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Properties for certain subclasses of analytic functions with nonzero coefficients

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.

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

A function fA is said to be in the class t (A,B,α) if

| f ( z ) 1 t ( A B ) B ( f ( z ) 1 ) |<α(zU),
(1.2)

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

In particular, for some real numbers A and B with 1B<A1 and α=1 without any restriction of the coefficients a n (nN={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 fA is said to be in the class UST(α) if

{ f ( z ) f ( ξ ) ( z ξ ) f ( z ) } >α(z×ξU×U;0α<1).

Furthermore, a function fA is said to be in the class UCV(α) if

{ 1 + ( z ξ ) f ( z ) f ( z ) } >α(z×ξU×U;0α<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,cC;c0,1,2,;zU),

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 {cab}>0, and it has a pole at z=1 if {cab}0. Moreover, univalence, starlikeness and convexity properties of zF(a,b;c;z) have been extensively studied by Ponnusamy and Vuorinen [13] and Ruscheweyh and Singh [14].

For fA, we define the operator I a , b ; c f by

I a , b ; c f(z)=zF(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 (nN{1}), then

n = 2 n ( 1 α | B | ) | a n |α|t||AB|.
(2.1)

Proof From the definition of t (A,B,α), we have

| f (z)1|<α|t(AB)B ( f ( z ) 1 ) |(zU),

and so

| n = 2 n a n z n 1 |<α|t(AB)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 (0r<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||AB|.
(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||AB|,
(2.5)

where A and B are complex numbers with AB, tC{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(AB)B ( f ( z ) 1 ) |(zU).
(2.6)

From (2.6), we have

| n = 2 n a n z n 1 |<α|t(AB)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||AB|.
(2.7)

Letting r 1 in (2.7), we have

n = 2 n ( 1 + α | B | ) | a n |α|t||AB|.

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 |1α(0α<1),
(2.8)

then fUST(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|1α ( 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 1α 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(2n1α)| a n |1α(0α<1),
(2.9)

then fUCV(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 ) |<1α ( 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 1α if the assumption (2.9) is satisfied. Thus we complete the proof of Theorem 4. □

Theorem 5 Let a,bC{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||AB|,

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 (0r<1) at a point z 0 , we can write

z 0 ω ( z 0 )=kω( z 0 )(k1).

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 (zU)
(2.10)

for some real α (0α<1), β>0, and γ>0. Then

| ( I a , b ; c f ( z ) ) 1|<1α(zU).
(2.11)

Proof Let us define ω by

ω(z)= ( I a , b ; c f ( z ) ) 1 1 α (zU).

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 ) =k1.

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(zU).

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(zU).

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

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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).

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Lee, H.J., Cho, N.E. & Owa, S. Properties for certain subclasses of analytic functions with nonzero coefficients. J Inequal Appl 2014, 506 (2014). https://doi.org/10.1186/1029-242X-2014-506

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