Coecient bounds for certain subclasses of analytic functions of complex order

Transcription

1 Hacettepe Journal of Mathematics and Statistics Volume 45 (4) (2016), Coecient bounds for certain subclasses of analytic functions of complex order Serap Bulut Abstract In this paper, we introduce and investigate two subclasses of analytic functions of complex order, which are introduced here by means of a certain nonhomogeneous CauchyEuler-type dierential equation of order m. Several corollaries and consequences of the main results are also considered. Keywords: Analytic functions, Dierential operator, Nonhomogeneous Cauchy- Euler dierential equations, Coecient bounds, Subordination AMS Classication: 30C45 Received : Accepted : Doi : /HJMS Introduction, denitions and preliminaries Let R = (, ) be the set of real numbers, C be the set of complex numbers, N := {1, 2, 3,...} = N 0\ {0} be the set of positive integers and N := N\ {1} = {2, 3, 4,...}. Let A denote the class of functions of the form (1.1) f(z) = z + a iz i i=2 which are analytic in the open unit disk U = {z : z C and z < 1}. Kocaeli University, Faculty of Aviation and Space Sciences, Arslanbey Campus, Kartepe-Kocaeli, TURKEY serap.bulut@kocaeli.edu.tr

2 1016 (1.2) Recently, Faisal and Darus [8] dened the following dierential operator: D 0 f (z) = f (z), ( D 1 α µ + β (α, β, µ) f (z) = α + β D 2 (α, β, µ) f (z) = D ( D 1 (α, β, µ) f (z) ). ) f (z) + D n (α, β, µ) f (z) = D ( D n 1 (α, β, µ) f (z) ). If f is given by (1.1), then it is easily seen from (1.2) that ( ) n α + (µ + ) (i 1) + β (1.3) D n (α, β, µ) f (z) = z + a iz i α + β i=2 (f A; α, β, µ, 0; α + β 0; n N 0). ( ) µ + zf (z), α + β By using the operator D n (α, β, µ), Faisal and Darus [8] dened a function class Ψ (n, α, β, µ,, ζ, γ, ξ) by { ( R [ z ζd (α, β, µ) f (z) + (1 ζ) D n (α, β, µ) f (z) ] )} ξ ζd (α, β, µ) f (z) + (1 ζ) D n (α, β, µ) f (z) 1 > γ, (z U; 0 γ < 1; 0 ζ 1; ξ C\ {0}) and also investigated the subclass Φ (n, α, β, µ,, ζ, γ, ξ, τ) of the analytic function class A, which consists of functions f A satisfying the following nonhomogenous Cauchy- Euler dierential equation: z 2 d 2 w dz 2 dw + 2 (1 + τ) z + τ (1 + τ) w = (1 + τ) (2 + τ) q(z) dz (w = f(z) A; q Ψ (n, α, β, µ,, ζ, γ, ξ) ; τ ( 1, )). In the same paper [8], coecient bounds for the subclasses Ψ (n, α, β, µ,, ζ, γ, ξ) and Φ (n, α, β, µ,, ζ, γ, ξ, τ) of analytic functions of complex order were obtained. Making use of the dierential operator D n (α, β, µ), we now introduce each of the following subclasses of analytic functions. 1. Denition. Let g : U C be a convex function such that g(0) = 1 and R {g (z)} > 0 (z U). We denote by M g (n, α, β, µ,, ζ, ξ) the class of functions f A satisfying ( [ z ζd (α, β, µ) f (z) + (1 ζ) D n (α, β, µ) f (z) ] ) ξ ζd (α, β, µ) f (z) + (1 ζ) D n (α, β, µ) f (z) 1 g (U), where z U; 0 ζ 1; ξ C\ {0}. 2. Denition. A function f A is said to be in the class M g (n, α, β, µ,, ζ, ξ; m, τ) if it satises the following nonhomogenous Cauchy-Euler dierential equation: ( ) ( ) z m d m w dz + m (τ + m 1) z m 1 d m 1 m 1 w m m w (τ + j) = q(z) (τ + j + 1) m 1 dzm 1 m (1.4) (w = f(z) A; q M g (n, α, β, µ,, ζ, ξ) ; m N ; τ ( 1, )).

3 1017 Remark 1. There are many choices of the function g which would provide interesting subclasses of analytic functions of complex order. In particular, (i) if we choose the function g as g (z) = 1 + Az ( 1 B < A 1; z U), 1 + Bz it is easy to verify that g is a convex function in U and satises the hypotheses of Denition 1. If f M g (n, α, β, µ,, ζ, ξ), then we have ( [ z ζd (α, β, µ) f (z) + (1 ζ) D n (α, β, µ) f (z) ] ξ ζd (α, β, µ) f (z) + (1 ζ) D n (α, β, µ) f (z) 1 ) 1 + Az 1 + Bz (z U). We denote this new class by H (n, α, β, µ,, ζ, ξ, A, B). Also we denote by B (n, α, β, µ,, ζ, ξ, A, B; m, τ) for corresponding class to M g (n, α, β, µ,, ζ, ξ; m, τ); (ii) if we choose the function g as 1 + (1 2γ) z g (z) = (0 γ < 1; z U), 1 z it is easy to verify that g is a convex function in U and satises the hypotheses of Denition 1. If f M g (n, α, β, µ,, ζ, ξ), then we have { R that is ξ ( [ z ζd (α, β, µ) f (z) + (1 ζ) D n (α, β, µ) f (z) ] ζd (α, β, µ) f (z) + (1 ζ) D n (α, β, µ) f (z) 1 f Ψ (n, α, β, µ,, ζ, γ, ξ). Remark 2. In view of Remark 1(ii), by taking 1 + (1 2γ) z g (z) = (0 γ < 1; z U) 1 z in Denitions 1 and 2, we easily observe that the function classes M g (n, α, β, µ,, ζ, ξ) and M g (n, α, β, µ,, ζ, ξ; 2, τ) become the aforementioned function classes Ψ (n, α, β, µ,, ζ, γ, ξ) and Φ (n, α, β, µ,, ζ, γ, ξ, τ), )} > γ (z U), respectively. In this work, by using the principle of subordination, we obtain coecient bounds for functions in the subclasses M g (n, α, β, µ,, ζ, ξ) and M g (n, α, β, µ,, ζ, ξ; m, τ) of analytic functions of complex order, which we have introduced here. Our results would unify and extend the corresponding results obtained earlier by Robertson [13], Nasr and Aouf [12], Altnta³ et al. [1], Faisal and Darus [8], Srivastava et al. [16], and others. In our investigation, we shall make use of the principle of subordination between analytic functions, which is explained in Denition 3 below (see [11]). 3. Denition. For two functions f and g, analytic in U, we say that the function f (z) is subordinate to g (z) in U, and write f (z) g (z) (z U), if there exists a Schwarz function w (z), analytic in U, with w (0) = 0 and w (z) < 1 (z U),

4 1018 such that f (z) = g (w (z)) (z U). In particular, if the function g is univalent in U, the above subordination is equivalent to f (0) = g (0) and f (U) g (U). 2. Main results and their demonstration In order to prove our main results (Theorems 1 and 2 below), we rst recall the following lemma due to Rogosinski [14]. 1. Lemma. Let the function g given by g (z) = b k z k (z U) k=1 be convex in U. Also let the function f given by f(z) = a k z k (z U) k=1 be holomorphic in U. If then f (z) g (z) (z U), a k b 1 (k N). We now state and prove each of our main results given by Theorems 1 and 2 below. 1. Theorem. Let the function f A be dened by (1.1). If the function f is in the class M g (n, α, β, µ,, ζ, ξ), then (α + β) (2.1) (i 1)! [α + ζ (µ + ) (i 1) + β] [α + (µ + ) (i 1) + β] n Proof. Let the function f A be given by (1.1). Suppose that the function F (z) is dened, in terms of the dierential operator D n (α, β, µ), by (2.2) F (z) = ζd (α, β, µ) f (z) + (1 ζ) D n (α, β, µ) f (z) (z U). Then, clearly, F is an analytic function in U, and a simple computation shows that F has the following power series expansion: (2.3) F (z) = z + A iz i (z U), i=2 where, for convenience, (2.4) A i = [α + ζ (µ + ) (i 1) + β] [α + (µ + ) (i 1) + β]n (α + β) a i From Denition 1 and (2.2), we thus have ( ) zf (z) ξ F (z) 1 g (U) (z U). Let us dene the function p(z) by (2.5) p(z) = ( ) zf (z) ξ F (z) 1 (z U).

5 1019 Hence we deduce that p(0) = g(0) = 1 and p(z) g (U) (z U). Therefore, we have p(z) g(z) (z U). Thus, according to the Lemma 1, we obtain (2.6) p(l) (0) l! g (0) (l N). Also from (2.5), we nd (2.7) zf (z) = [1 + ξ (p(z) 1)] F (z). Next, we suppose that (2.8) p(z) = 1 + c 1z + c 2z 2 + (z U). Since A 1 = 1, in view of (2.3), (2.7) and (2.8), we obtain (2.9) (i 1) A i = ξ {c i 1 + c A c 1A i 1} By combining (2.6) and (2.9), for i = 2, 3, 4, we obtain A 2 ξ g (0), A 3 ξ g (0) (1 + ξ g (0) ), 2! A 4 ξ g (0) (1 + ξ g (0) ) (2 + ξ g (0) ), 3! respectively. Also, by using the principle of mathematical induction, we obtain A i (i 1)! Now from (2.4), it is clear that (α + β) (i 1)! [α + ζ (µ + ) (i 1) + β] [α + (µ + ) (i 1) + β] n This evidently completes the proof of Theorem Theorem. Let the function f A be dened by (1.1). If the function f is in the class M g (n, α, β, µ,, ζ, ξ; m, τ), then (α + β) m 1 (τ + j + 1) (i 1)! [α + ζ (µ + ) (i 1) + β] [α + (µ + ) (i 1) + β] n m 1 (τ + j + i) Proof. Let the function f A be given by (1.1). Also let h(z) = z + b iz i M g (n, α, β, µ,, ζ, ξ). i=2

6 1020 Hence, from (1.4), we deduce that a i = m 1 m 1 (τ + j + 1) b i (i N, τ ( 1, )). (τ + j + i) Thus, by using Theorem 1, we obtain (α + β) (i 1)! [α + ζ (µ + ) (i 1) + β] [α + (µ + ) (i 1) + β] n This completes the proof of Theorem Corollaries and consequences m 1 m 1 (τ + j + 1). (τ + j + i) In this section, we apply our main results (Theorems 1 and 2 of Section 2) in order to deduce each of the following corollaries and consequences. 1. Corollary. ([19]) Let the function f A be dened by (1.1). If the function f is in the class M g (0, α, β, µ,, ζ, ξ) S g(ζ, ξ), then (i 1)! (1 + ζ (i 1)) 2. Corollary. ([19]) Let the function f A be dened by (1.1). If the function f is in the class M g (0, α, β, µ,, ζ, ξ; m, τ) K g(ζ, ξ, m; τ), then (i 1)! (1 + ζ (i 1)) m 1 m 1 (τ + j + 1) (τ + j + i) 3. Corollary. ([17]) Let the function f A be dened by (1.1). If the function f is in the class M g (n, 1, 0, 0, 1, ζ, ξ) M g(n, ζ, ξ), then i n (1 + ζ (i 1)) (i 1)! 4. Corollary. ([17]) Let the function f A be dened by (1.1). If the function f is in the class M g (n, 1, 0, 0, 1, ζ, ξ; 2, τ) M g(n, ζ, ξ; τ), then Setting (1 + τ) (2 + τ) i n (1 + ζ (i 1)) (i 1)! (i + τ) (i τ) m = 2 and g (z) = 1 + (1 2γ) z 1 z (0 γ < 1; z U) in Theorems 1 and 2, we have following corollaries, respectively.

7 Corollary. [8] Let the function f A be dened by (1.1). If the function f is in the class Ψ (n, α, β, µ,, ζ, γ, ξ), then (α + β) [j + 2 ξ (1 γ)] (i 1)! [α + ζ (µ + ) (i 1) + β] [α + (µ + ) (i 1) + β] n 6. Corollary. [8] Let the function f A be dened by (1.1). If the function f is in the class Φ (n, α, β, µ,, ζ, γ, ξ, τ), then (1 + τ) (2 + τ) (α + β) [j + 2 ξ (1 γ)] (i + τ) (i τ) (i 1)! [α + ζ (µ + ) (i 1) + β] [α + (µ + ) (i 1) + β] n For several other closely-related investigations, see (for example) the recent works [1-7, 12, 13]. References [1] Altnta³, O., Irmak, H., Owa, S. and Srivastava, H.M. Coecient bounds for some families of starlike and convex functions of complex order, Appl. Math. Letters 20, , [2] Altnta³, O., Irmak, H. and Srivastava, H.M. Fractional calculus and certain starlike functions with negative coecients, Comput. Math. Appl. 30 (2), 9-15, [3] Altnta³, O. and Özkan, Ö. Starlike, convex and close-to-convex functions of complex order, Hacet. Bull. Nat. Sci. Eng. Ser. B 28, 37-46, [4] Altnta³, O. and Özkan, Ö. On the classes of starlike and convex functions of complex order, Hacet. Bull. Nat. Sci. Eng. Ser. B 30, 63-68, [5] Altnta³, O., Özkan, Ö. and Srivastava, H.M. Neighborhoods of a class of analytic functions with negative coecients, Appl. Math. Letters 13 (3), 63-67, [6] Altnta³, O., Özkan, Ö. and Srivastava, H.M. Majorization by starlike functions of complex order, Complex Variables Theory Appl. 46, , [7] Altnta³, O., Özkan, Ö. and Srivastava, H.M. Neighborhoods of a certain family of multivalent functions with negative coecient, Comput. Math. Appl. 47, , [8] Faisal, I. and Darus, M. Application of nonhomogenous Cauchy-Euler dierential equation for certain class of analytic functions, Hacet. J. Math. Stat. 43 (3), , [9] Flett, T.M. The dual of an inequality of Hardy and Littlewood and some related inequalities, J. Math. Anal. Appl. 38, , [10] Jurg, I.B., Kim, Y.C. and Srivastava, H.M. The Hardy space of analytic functions associated with certain one-parameter families of integral operators, J. Math. Anal. Appl. 176, , [11] Miller, S.S. and Mocanu, P.T. Dierential subordinations: Theory and applications. Monographs and Textbooks in Pure and Applied Mathematics, 225. Marcel Dekker, Inc., New York, [12] Nasr, M.A. and Aouf, M.K. Radius of convexity for the class of starlike functions of complex order, Bull. Fac. Sci. Assiut Univ. A 12 (1), , [13] Robertson, M.S. On the theory of univalent functions, Ann. of Math. (2) 37 (2), , [14] Rogosinski, W. On the coecients of subordinate functions, Proc. London Math. Soc. (Ser. 2) 48, 48-82, [15] S l gean, G.. Subclasses of univalent functions, Complex Analysis-Fifth Romanian-Finnish seminar, Part 1 (Bucharest, 1981), Lecture Notes in Math., vol. 1013, Springer, Berlin, 1983, pp [16] Srivastava, H.M., Altnta³, O. and Serenbay, S.K., Coecient bounds for certain subclasses of starlike functions of complex order, Appl. Math. Letters 24 (8), , [17] Srivastava, H.M., Xu, Q.-H. and Wu, G.-P. Coecient estimates for certain subclasses of spiral-like functions of complex order, Appl. Math. Letters 23, , 2010.

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