Malaya J. Mat. 4(1)(2016) 37-41

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1 Malaya J. Mat. 4(1)(2016) Certain coefficient inequalities for -valent functions Rahim Kargar a,, Ali Ebadian a and Janus Sokół b a Deartment of Mathematics, Payame Noor University, I. R. of Iran. b Deartment of Mathematics, Resów University of Technology, Al. Powstańców Warsawy 12, Resów, Poland. Abstract In the resent aer, alying lemmas due to Nunokawa et al. [3] and Jack s lemma we obtain some coefficient inequalities for certain subclass of -valent functions. Keywords: Analytic, univalent, -valent, starlike and convex functions, Jack s lemma MSC: 30C45. c 2012 MJM. All rights reserved. 1 Introduction Let A denote the class of functions f of the form: f = + n=+1 a n n, N := 1, 2, 3,...}, (1.1) which are analytic in the oen unit disk = C : < 1}. Put A 1 = A. The subclass of A consisting of all univalent functions f in is denoted by S. A function f S is called starlike (with resect to 0), denoted by f S, if tw f ( ) whenever w f ( ) and t [0, 1]. A function f S that mas onto a convex domain, denoted by f K, is called a convex function. A function f in A is said to be starlike of order 0 γ < 1 if it satisfies > γ,. f We denote by S (γ) the subclass of A consisting of all starlike functions of order γ in. Furthermore, let M(β) be the class of functions f A which satisfy < β,. f for some real number β with β > 1. The class M(β) was investigated by Uralegaddi, Ganigi and Sarangi [6]. Further, let P(γ, ) denote the subclass of A consisting of functions f which satisfy } f > γ,, for some real 0 γ <. The class P(1/2, 1) P(1/2) was studied by Obradović et al. in [5]. We remark that K P(1/2). Nunokawa, Cho, Kwon and Sokół [3] obtained the following results. Corresonding author. address: rkargar1983@gmail.com (Rahim Kargar), ebadian.ali@gmail.com(ali Ebadian), jsokol@r.edu.l (Janus Sokół).

2 38 Rahim Kargar et al. / Certain coefficient inequalities for -valent functions Lemma 1.1. Let B and C be analytic in with ImC} < B}. If is analytic in with (0) = 1, and if argb + C} < π/2 + t, where we have argc + ib} when argc + ib} [0, π/2] t = argc + ib} π/2 when argc + ib} (π/2, π], } > 0,. Lemma 1.2. Let B and C be analytic in with } C 1,. B If is analytic in with (0) = 0, and if B + C < B + C,, (1.2) we have < 1,. In this aer, alying the Lemma 1.1, Lemma 1.2 and Jack s Lemma, we obtain coefficient conditions for some certain subclasses of -valent functions. 2 Main results Our first result is contained in the following: Theorem 2.1. Assume that f A. If arg 1 f ( 1) f γ } < 3π, 4, (2.3) where 0 γ <, that is f P(γ/, ). } f > γ,, (2.4) Proof. Let f = 0 for = 0 and let be defined by ( 1 γ ) + γ = f,, (2.5) where 0 γ <. Then is analytic in, (0) = 1 and ( 1 γ ) ( + 1 γ ) = 1 f ( 1) f γ. If we ut B = C = 1 γ, from (2.3) and alying Lemma 1.1 we obtain (2.4) immediately. If we take = 1 in Theorem 2.1, it becomes the result from [4] of the following form: Corollary 2.1. Let f A. If arg f γ} < 3π 4,, } f > γ,.

3 Rahim Kargar et al. / Certain coefficient inequalities for -valent functions 39 Theorem 2.2. Assume that f A. If f P(γ/, ), where 0 γ <. 1 f ( 1) f γ (1 < 2 γ ),, (2.6) Proof. For 0 γ <, let be defined by (2.5). Then from (2.6) and alying Lemma 1.2, we can obtain the result. Putting = 1, in Theorem 2.2, we have: Corollary 2.2. Let f A. If f γ < 2(1 γ),, } f > γ,. Putting γ = 1/2, in Corollary 2.2, we have: Corollary 2.3. Let f A. If f 1 2 < 1,, f P(1/2). The following Lemma (oularly known Jack s lemma (see [1])) will be required on our resent investigation. Lemma 2.3. Let the (nonconstant) function w be analytic in with w(0) = 0. If w attains its maximum value on the circle = r < 1 at a the oint 0, where c is a real number and c 1. 0 w ( 0 ) = cw( 0 ), Theorem 2.3. Assume that f / = γ and that f A satisfies the inequality f P( 1+γ 2, ), where 0 γ <. Proof. Define the function w by > + γ 1,, (2.7) f 2(γ + 1) f = 1 + γw, (w = 1, < 1), (2.8) 1 + w where 0 γ <. Because f / = γ, w is analytic in and w(0) = 0. From (2.7), some comutation yields f f = + γw 1 + γw w 1 + w. (2.9) Suose there exists a oint 0 such that Alying Lemma 2.3, we have w( 0 ) = 1 and w < 1 when < 0. 0 w ( 0 ) = cw( 0 ) (c 1, w( 0 ) = e, θ R). (2.10)

4 40 Rahim Kargar et al. / Certain coefficient inequalities for -valent functions Thus, by using (2.9) and (2.10), it follows that 0 f } ( 0 ) cγe = + f ( 0 ) 1 + γe cγ(γ + cos θ) = γ 2 + 2γ cos θ c 2 (2 + 1)γ + (2 1), 2(1 + γ) } γe } 1 + e which contradicts the hyothesis (2.7). It follows that w < 1, that is, ( f / ) 1 γ ( f / ) < 1, (, 0 γ < ). This evidently comletes the roof of Theorem 2.3. If we take γ = 0 and = 1, in Theorem 2.3, we get: Corollary 2.4. Let f A. If f P(1/2), that is S (1/2) P(1/2). > 1 f 2,, Theorem 2.4. Assume that f A satisfies the inequality < + 1 γ, f 2 γ, (2.11) where 0 γ <. f 1 < 1 γ,, Proof. Let us f / = γ. Consider the function w defined by f = 1 + (1 γ)w, < 1, (2.12) where 0 γ <. Then w is analytic in and w(0) = 0. From (2.12), some comutation yields Suose there exists a oint 0 such that Alying Lemma 2.3, we have f f = + (1 γ)w 1 + (1 γ)w. (2.13) w( 0 ) = 1 and w < 1 when < 0. 0 w ( 0 ) = cw( 0 ) (c 1, w( 0 ) = e, θ R). (2.14) Thus, by using (2.13) and (2.14), it follows that 0 f } ( 0 ) c(1 γ)e } = + f ( 0 ) 1 + (1 γ)e c(1 γ)(1 γ + cos θ) = (1 γ) 2 + 2(1 γ) cos θ (2 + 1) γ( + 1), 2 γ which contradicts the hyothesis (2.11). It follows that w < 1, that is, f 1 < 1 γ, (, 0 γ < ). This evidently comletes the roof of Theorem 2.4.

5 Rahim Kargar et al. / Certain coefficient inequalities for -valent functions 41 Corollary 2.5. Assume that f A. If f satisfies the inequalities < γ f 2 γ,, Taking γ = 1/2 in Corollary 2.5, we have: } f > γ,, 0 γ < 1. Corollary 2.6. Assume that f A. If f satisfies the inequalities < 4 f 3,, that is, M(4/3) P(1/2). f P(1/2),, Combining Corollary 2.4 and 2.6, we have: Corollary 2.7. Assume that f A. If f satisfies the following two-sided inequality 1 f 2 < } < 4 f 3,, f P(1/2),, that is, S(1/2, 3/4) P(1/2), where the class S(α, β), α < 1 and β > 1, was recently considered by K. Kuroki and S. Owa in [2]. ferences [1] I. S. Jack, Functions starlike and convex of order alha, J. London Math. Soc. 3(2)(1971), [2] K. Kuroki and S. Owa, New class for certain analytic functions concerned with the stri domains, Advances in Mathematics: Scientific Journal 2(2)(2013), [3] M. Nunokawa, N. E. Cho, O. S. Kwon and J. Sokół, On differential subordinations in the comlex lane, Journal of Inequalities and Alications, (2015) 2015:7. [4] M. Nunokawa, S. Owa, K. Kuroki and J. Sokół, Some generaliition roerties of analytic functions concerned with Sakaguchi result, Tamkang Journal of Mathematics, 46(2)(2015), [5] M. Obradović and S. Ponuusamy and N. Tuneski, Radius of univalence of certain combination of univalent of analytic functions, Bull. Malays. Math. Sci. Soc., 35(2)(2012), [6] B. A. Uralegaddi, M. D. Ganigi, S. M. Sarangi, Univalent functions with ositive coefficients, Tamkang J. Math., 25(1994), ceived: June 28, 2015; Acceted: August 23, 2015 UNIVERSITY PRESS Website: htt://