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Abstract
In this paper we introduce and investigate an interesting subclass LBh;p
˙
./ of ana-
lytic and bi-univalent functions in the open unit disk U. For functions belonging to the class
LBh;p
˙
./, we obtain estimates on the first two Taylor-Maclaurin coefficients a2 and a3. The
results presented in this paper would generalize and improve some recent work of Joshi et al. [5].
2010 Mathematics Subject Classification: 30C45
Results & Lemmas (6)
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Theorem 1
Theorem 1 ([5]). Let f.´/ given by.1:1/ be in the class LB ˙.˛/.0 < ˛ 1; 1/. Then ja2j 2˛ p.2 1/.2 1C˛/ (1.5)
Theorem 1 ([5]). Let f .´/ given by .1:1/ be in the class LB ˙ .˛/ .0 < ˛ 1; 1/. Then ja2j 2˛ p .2 1/.2 1C˛/ (1.5)
Theorem 2
Theorem 2 ([5]). Let f.´/ given by.1:1/ be in the class LB˙.;ˇ/.0 ˇ < 1; 1/. Then ja2j s 2.1 ˇ/ .2 1/ (1.9) and ja3j …
Theorem 2 ([5]). Let f .´/ given by .1:1/ be in the class LB˙ .;ˇ/ .0 ˇ < 1; 1/. Then ja2j s 2.1 ˇ/ .2 1/ (1.9) and ja3j 4.1 ˇ/2 .2 1/2 C 2.1 ˇ/ 3 1 : (1.10) Here, in our present paper, inspiring by some of the aforecited works (especially [5]), we introduce the following subclass of the analytic function class A, analog- ously to the definition given by Xu et al. [7].
Theorem 3.
Theorem 3. Let the function f.´/ given by the Taylor-Maclaurin series expansion.1:1/ be in the bi-univalent function class LBh;p ˙./: Then…
Theorem 3. Let the function f .´/ given by the Taylor-Maclaurin series expansion .1:1/ be in the bi-univalent function class LBh;p ˙ ./: Then ja2j min 8 < : s jh0 .0/j2 Cjp0 .0/j2 2.2 1/2 ; s jh00 .0/jCjp00 .0/j 4.2 1/ 9
Corollary 1
Corollary 1 ([3]). Let the function f.´/ given by the Taylor-Maclaurin series expansion.1:1/ be in the bi-univalent function class Bh;p ˙:…
Corollary 1 ([3]). Let the function f .´/ given by the Taylor-Maclaurin series expansion .1:1/ be in the bi-univalent function class Bh;p ˙ : Then ja2j min 8 < : s jh0 .0/j2 Cjp0 .0/j2 2 ; r jh00 .0/jCjp00 .0/j 4 9
Corollary 2.
Corollary 2. Let the function f.´/ given by the Taylor-Maclaurin series expan- sion.1:1/ be in the bi-univalent function class LB ˙.˛/.0 <…
Corollary 2. Let the function f .´/ given by the Taylor-Maclaurin series expan- sion .1:1/ be in the bi-univalent function class LB ˙ .˛/ .0 < ˛ 1; 1/. Then ja2j s 2˛2 .2 1/ and ja3j 8 ˆ< ˆ: 2˛2 .2 1/ ;
Corollary 3.
Corollary 3. Let the function f.´/ given by the Taylor-Maclaurin series expan- sion.1:1/ be in the bi-univalent function class LB˙.;ˇ/.0 …
Corollary 3. Let the function f .´/ given by the Taylor-Maclaurin series expan- sion .1:1/ be in the bi-univalent function class LB˙ .;ˇ/ .0 ˇ < 1; 1/. Then ja2j 8 ˆ< ˆ: q 1 ˇ .2 1/ 0 ˇ 2C1 4 2.1 ˇ/ .2 1/ 2C1 4
Function classes studied:
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