Some Starlikeness Criterions for Analytic Functions

Let be a positive integer, and let denote the class of function

(1.1)

that are analytic in the unit disk . For , let

(1.2)

denote the class of starlike function of order and .

Let and be analytic in ; then we say that the function is subordinate to in , if there exists an analytic function in such that , and , denoted that or . If is univalent in , then the subordination is equivalent to and [1].

Let

(1.3)

Singh [2] proved that if . More recently, Fournier [3, 4] proved that

(1.4)

is the order of starlikeness of . Now, we define

(1.5)

Clearly, . In 1998, Obradović [5] proved that

(1.6)

if and . Recently, Obradović and Owa [6] proved that

(1.7)

if and .

In this paper we find a condition on , , , and for which

(1.8)

implies and extend some results of Obradović and Owa [5, 6]. Also, we obtain some new results on starlikeness criterions.

For our results we need the following lemma.

Lemma 2.1 (see [6]).

Let be analytic in , , and satisfy the condition

(2.1)

Then

(2.2)

where

(2.3)

Theorem 2.2.

Let , , , and

(2.4)

where

(2.5)

If and are analytic in , satisfy

(2.6)

(2.7)

where , then

(2.8)

Proof.

If , it is easy to see the result is true. Now, assume . Let

(2.9)

If there exists , such that , then we will show that

(2.10)

for . Note that for ; it is sufficient to show that

(2.11)

for . Let , ; then, the left-hand side of (2.11) is

(2.12)

Suppose that and note that ; then inequality (2.11) is equivalent to

(2.13)

for all and . Now, if we define

(2.14)

then we have

(2.15)

where

(2.16)

Since

(2.17)

the denominator of is positive. Further, let

(2.18)

We have

(2.19)

If

(2.20)

we get

(2.21)

where

(2.22)

Note that

(2.23)

We obtain

(2.24)

If

(2.25)

we have

(2.26)

Hence we obtain

(2.27)

where

(2.28)

If

(2.29)

we have . It follows that .

Therefore we obtain for if or . It follows that

(2.30)

If , we have

(2.31)

by (2.13) and (2.21) for and by (2.23) for . Note that is an continuous increasing function for , and

(2.32)

Then there exists a unique , such that

(2.33)

Thus, is the global minimum point of on [ . It follows from (2.33) that

(2.34)

or

(2.35)

By a simple calculation, we may obtain

(2.36)

for . It follows from (2.30) and (2.36) that that inequality (2.13) holds. This shows that inequality (2.10) holds, which contradicts with (2.7). Hence we must have

(2.37)

Theorem 2.3.

Let , , , and be defined as in Theorem 2.2. If satisfies

(2.38)

where , then .

Proof.

If , and the result is trivial. Now, assume . If we put

(2.39)

then by some transformations and (2.38) we get

(2.40)

By Lemma 2.1, we obtain

(2.41)

Let

(2.42)

Then we have

(2.43)

By Theorem 2.2, we get

(2.44)

It follows that .

For , we get the following corollary.

Corollary 2.4.

Let , , and let

(2.45)

where

(2.46)

If satisfies

(2.47)

where , then .

Corollary 2.5.

Let , , and let

(2.48)

If satisfies

(2.49)

where , then .

Proof.

Note that

(2.50)

Putting in Theorem 2.3, we obtain the above corollary.

Remark 2.6.

Our results extend the results given by Obradović [5], and Obradović and Owa [6].

Theorem 2.7.

Let , , , and let

(2.51)

If satisfies

(2.52)

where , and

(2.53)

then .

Proof.

Let

(2.54)

Then from (2.52) and (2.53) we obtain

(2.55)

Hence, by using Theorem 1 given by Hallenbeck and Ruscheweyh [7], we have that

(2.56)

and the desired result easily follows from Corollary 2.5.

For , we have the following corollary.

Corollary 2.8.

Let , , and let

(2.57)

If satisfies

(2.58)

where , and

(2.59)

then .