NOTE ON RADIUS PROBLEMS FOR CERTAIN CLASS OF ANALYTIC FUNCTIONS. We dedicate this paper to the 60th anniversary of Professor Y. Polatoglu.

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1 NOTE ON RADIUS PROBLEMS FOR CERTAIN CLASS OF ANALYTIC FUNCTIONS Neslihan Uyanik 1, Shigeyoshi Owa 2 We dedicate this paper to the 60th anniversary of Professor Y. Polatoglu Abstract For analytic functions f) normalied by f0) = f 0) 1 = 0 in the open unit disk U, the subclass Aα, β; λ) of functions f) which satisfy ) 1 α + β 2 f) f) 1 ) λ U) for some complex numbers α and β and for some real λ > 0 is introduced. The object of the present paper is to discuss some radius properties for S γ) such that 1 f) Aα, β; λ). MSC 2010: 30C45 Keywords and Phrases: analytic function, starlike function of order α, Cauchy-Schwar inequality 1. Introduction Let A be the class of functions f) of the form f) = + a n n 1.1) c 2010, FCAA Diogenes Co. Bulgaria). All rights reserved.

2 554 N. Uyanik, S. Owa which are analytic in the open unit disk U = { C : < 1}. For a function f) A, we say that f) belongs to the class Aα, β; λ) if it satisfies f) 0 U) and ) 1 α + β 2 f) f) 1 ) λ U) 1.2) for some complex numbers α and β and for some real λ > 0. Let us consider a function f k ) given by f k ) = 1 ) k k R). Then f k ) satisfies f k) = 1 0 U) and 1 ) k ) = kk 1)1 ) k 2. f k ) Further, if we write that with f k ) = 1 + a n n ) a n = 1) n k, n then we see that 1 2 f k ) 1 ) = n 1)a n n. Therefore, we have that ) 1 α +β 2 f k ) f k ) 1 ) = αkk 1)1 )k 2 +β < α kk 1)2 k 2 + β n 1) a n n 1)a n n

3 NOTE ON RADIUS PROBLEMS FOR CERTAIN CLASS for k 2. This means that f k ) Aα, β; λ) for λ 2 α + β if k = 2, and f k ) Aα, β; λ) for λ 12 α + 5 β if k = 3. The classes A1, 0; λ) and A0, 1; λ) were introduced by Obradović and Ponnusamy [2], the generalied classes of A1, 0; λ) and A0, 1; λ) were considered by Shimoda, Hayami, Hashidume and Owa [4] and Kobashi, Kuroki, Shiraishi and Owa [1]. Let S γ) denote the subclass of A consisting of all functions f) which satisfy f ) ) 1.3) Re > γ U) f) for some real γ 0 γ < 1). A function f) S λ) is said to be starlike of order γ in U cf. Robertson [3]). We also write that S 0) = S. For f) A given by 1.1), we write f) = = 1 + a n n 1 b n n. Then we know that b 1 = a 2, b 2 = a 2 2 a 3 and b 3 = 2a 2 a 3 a 4 a Radius problems To discuss our radius problems for f) Aα, β; λ), we need the following lemmas. Lemma 2.1. Let f) A be given by 1.4) with f) 0 U). If f) satisfies n 1) α n + β ) b n λ 2.1) for some complex numbers α and β and for some real λ > 0, then f) Aα, β; λ). P r o o f. It follows that ) 1 α +β 2 f) f) 1 ) = α nn 1)b n n 2 +β n 1)b n n

4 556 N. Uyanik, S. Owa < α nn 1) b n + β n 1) b n = n 1) α n + β ) b n. Therefore, if the coefficient inequality 2.1) holds true, then we say that f) Aα, β; λ). Lemma 2.2. Let f) A be given by 1.4) with f) 0 U). Further let b n = b n e inθ θ R). If f) S γ), then n + γ 1) b n 1 γ. 2.2) P r o o f. Note that f) S γ) implies that f ) 1 n 1)b ) n n Re = Re f) 1 + b n n 1 n 1) b n e inθ n = Re 1 + > γ U). b n e inθ n n 1 If we consider such that = e iθ, then we obtain that 1 n 1) b n n 1 + > γ < 1). b n n Now, letting 1, we have that n + γ 1) b n 1 γ, Remark 2.1. In view of the coefficient inequality 2.2), we know that n + γ 1) b n 1 γ γ b 1 which shows that

5 NOTE ON RADIUS PROBLEMS FOR CERTAIN CLASS b n 1 γ γ b 1 < 1 n = 2, 3, 4, ). n + γ 1 This implies that 2.3) n 1) b n 2 1 γ γ b 1. Now, we derive the following Theorem 2.1. Let f) A be given by 1.4) with f) 0 U). Further let b n = b n e inθ θ R) and C < 1). If f) S γ) 1 with 0 γ < 1 + b 1, then 1 f) belongs to the class Aα, β; λ) for 0 < 0 λ), where 0 λ) is the smallest positive root of the equation 2 α ) + β 1 )) 2 1 γ γ b1 = λ1 2 ) ) P r o o f. By means of 1.4), we have that 1 f) = 1 + b n n n. Thus, we have to prove that n 1) α n + β ) b n n λ by Lemma 2.1. Applying the Cauchy-Schwar inequality, we see that n 1) α n + β ) b n n = α nn 1) b n n + β n 1) b n n ) 1 2 ) 1 2 α n 2 n 1) 2n n 1) b n 2 We note that ) 1 2 ) β n 1) 2n n 1) b n 2. ) 1 2 n 1) b n 2 1 γ γ b 1

6 558 N. Uyanik, S. Owa from 2.3) of Remark 2.1. If we put 2 = x, then n 2 n 1) 2n = x 2 n 2 n 1)x n 2 and ) ) = x 2 n n = x 2 x 1 x) 2 x = 2x2 2 + x) 1 x) 4 n 1) 2n = x 2 n 1)x n 2 ) ) x = x 2 x n 1 = x 2 = 1 x Therefore, we obtain that n 1) α n + β ) b n n x 2 1 x) 2. α 2 ) ) 1 2 ) 2 + β 2 1 γ γ b Considering λ > 0 such that α 2 ) ) 1 2 ) 2 + β 2 1 γ γ b1 1 2 = λ which is equivalent to 2.4), we define h ) by h ) = 2 α ) + β 1 2 )) 1 γ γ b1 λ1 2 ) 2. Then we have that h0) = λ < 0 and h1) = 6 α 1 γ γ b 1. This implies that h ) = 0 has a positive root 0 λ) for 0 < < 1. This completes the proof of the theorem. Taking γ = 0 in Theorem 2.1, we have the following Corollary 2.1. Let f) A be given by 1.4) with f) 0 U). Further, let b n = b n e inθ θ R) and C 0 < < 1). If f) S, then 1 f) belongs to the class Aα, β; λ) for 0 < 0λ), where 0 λ) is the smallest positive root of the equation 2 α ) ) + β 1 2 ) = λ1 2 ) )

7 NOTE ON RADIUS PROBLEMS FOR CERTAIN CLASS Remark 2.2. If we take = 1 2 eiθ in 2.5), then we have λ = α β. If we consider α = β = λ = 1, then ) ) 1 2 )2 = 0. Therefore, we see that < 0 1) < Taking α = 1 and β = 0 in Theorem 2.1, we have Corollary 2.2. Let f) A be given by 1.4) with f) 0 U). Further, let b n = b n e inθ θ R) and C 0 < < 1). If f) S γ) 1 with 0 γ < 1 + b 1, then 1 f) belongs to the class A1, 0; λ) for 0 < 0 λ), where 0 λ) is the smallest positive root of the equation ) 1 2 ) 2 1 γ γ b1 = λ. If we take α = 0 and β = 1 in Theorem 2.1, then we have Corollary 2.3. Let f) A be given by 1.4) with f) 0 U). Further let b n = b n e inθ θ R) and C 0 < < 1). If f) S γ) with 0 γ < b 1, then 1 f) belongs to the class A0, 1; λ) for 0 < 0 λ), where ) 1 0 λ) = 2 λ λ +. 1 γ γ b 1 Finally, since b 1 = 0 implies that a 2 = 0, we have Corollary 2.4. Let f) A be given by 1.4) with a 2 = 0 and f) 0 U). Further, let b n = b n e inθ θ R) and C < 1). If f) S γ) with 0 γ < 1, then 1 f) belongs to the class Aα, β; λ) for 0 < 0 λ), where 0 λ) is the smallest positive root of the equation 2 α ) + β 1 2 )) 1 γ = λ1 2 ) 2.

8 560 N. Uyanik, S. Owa References [1] H. Kobashi, K. Kuroki, H. Shiraishi and S. Owa, Radius problems of certain analytic functions. Internat. J. Open Problems Complex Anal ), [2] M. Obradovć and S. Ponnusamy, Radius properties for subclasses of univalent functions. Analysis ), [3] M.S. Robertson, On the theory of univalent functions. Ann. of Math ), [4] Y. Shimoda, T. Hayami, Y. Hashidume and S. Owa, Radius properties of certain analytic functions. Internat. J. Open Problems Complex Anal ), Department of Mathematics Kaim Karabekir Faculty of Education Atatürk University Erurum T-25240, TURKEY nesuyan@yahoo.com Received: GFTA, August 27-31, Departmemt of Mathematics Kinki University Higashi-Osaka, Osaka , JAPAN owa@math.kindai.ac.jp