TAIWANESE JOURNAL OF MATHEMATICS Vol. 15, No. 5, pp. 2377-2386, October 2011 This paper is available online at http:/tjm.math.ntu.edu.tw/index.php/TJMCOEFFICIENT ESTIMATES FOR CERTAIN SUBCLASSES OF ANALYTIC FUNCTIONS OF COMPLEX ORDERQing-Hua Xu, Ying-Chun Gui and H. M. Srivastava*Abstract. In this paper, we introduce and investigate each of the following subclasses:Sg(,) and Kg(,m;u)? 0?1;uR (,1; m N 1?of analytic functions of complex order C 0, g : U C being some suitably constrained convex function in the open unit disk U.We obtain coefficient bounds and coefficient estimates involving the Taylor-Maclaurin coefficients of the function f(z) when f(z) is in the class Sg(,) or in the class Kg(,m;u). The various results, which are presented in this paper, would generalize and improve those in related works of several earlier authors.1. INTRODUCTION, DEFINITIONS ANDPRELIMINARIESLet C be the set of complex numbers andN = 1,2,3, = N0 0be the set of positive integers. We also let A denote the class of functions of the form:f(z) = z +?n=2anzn,(1)which are analytic in the open unit diskU = z : z Cand|z| 0(z U; C:= C 0).Furthermore, a function f(z) A is said to be in the class C() of convex functions of complex order if it satisfies the following inequality:(3)? 1 +1 ?zf?(z)f?(z)? 0(z U; C).The function classesS()and C()were investigatedearlierby Nasr and Aouf 14 (see also 15) and Wiatrowski 20, respectively, and (more recently) by Altintas et al. (1 to 10), Deng 11, Murugusundaramoorthy and Srivastava 13, Srivastava et al. 19, and others (see, for example, 12 and 18). For two functions f and g, analytic in U, we say that f(z) is subordinate to g(z) in U and we write f g or, more precisely,f(z) g(z)(z U)if there exists a Schwarz function w(z), analytic in U withw(0) = 0and|w(z)| 0(z U). We denote by Sg(,) the class of functions given by(8)Sg(,)=? f :fA and 1+1 ?zf?(z)+z2f?(z) zf?(z)+(1 )f(z)1? g(U) (zU)?(0 ? ? 1; C).2380Qing-Hua Xu, Ying-Chun Gui and H. M. SrivastavaDefinition 4. A function f A is said to be in the class Kg(,m;u) if it satisfies the following nonhomogenous Cauchy-Euler differential equation:zmdmw dzm+?m1? (u + m 1)zm1dm1w dzm1+ +?mm? wm1?j=0(u + j)= h(z)m1?j=0(u+ j + 1)(9)?w = f(z) A; h(z) S g(,); u R (,1; m N?.Remark 1. There are many choices of the function g(z) which would provide interesting subclasses of analytic functions of complex order C. In particular, if we let(10)g(z) =1 + Az 1 + Bz(1 ? B A ? 1; z U),it is fairly easy to verify that g(z) is a convex function in U and satisfies the hypotheses of Definition 3. Clearly, therefore, the function class Sg(,), with the function g(z) given by (10), coincides with the function class S(,A,B) given by Definition 1.Remark 2. In view of Remark 1, if the function g(z) is given by (10), it is easily observed that the function classesSg(,)andKg(,m;u)reduce to the aforementioned function classesS(,A,B)andK(,A,B,m;u),respectively (see Definitions 1 and 2). In this paper, by using the subordination principle between analytic functions, we obtain coefficient bounds for the Taylor-Maclaurin coefficients for functions in the substantially more general function classesSg(,)andKg(,m;u)of analytic functions of complex order C. The various results presented here would generalize and improve the corresponding results obtained by (for example) Srivastava et al. 17.2. MAINRESULTS ANDTHEIRDERIVATIONSIn order to prove our main results, we will need the following lemma due to Rogosinski 16.Coefficient Estimates for Certain Subclasses of Analytic Functions of Complex Order2381Lemma (see 16). Let the function g(z) given byg(z) =?k=1bkzk(z U)be convex in U. Also let the function f(z) given byf(z) =?k=1akzk(z U)be holomorphic in U. Iff(z) g(z)(z U),then(11)|ak| ? |b1|(k N).Our first main result is now stated as Theorem 3 below.Theorem 3. Let the function f(z) be defined by (1). If f Sg(,), then(12)|an| ?n2?k=0(k + |g?(0)| |)(n 1)!1 + (n 1)(n N).Proof.Let the function F(z) be defined byF(z) = zf?(z) + (1 )f(z)(z U).Then, clearly, F(z) is an analytic functionin U, F(0) = 1,and a simplecomputation shows that the function F(z) has the following Taylor-Maclaurin series expansion:F(z) = z +?j=2Ajzj(z U),(13)where, for convenience,Aj= (1 + j)aj(j N).(14)Now, from Definition 3, we have1 +1 ?zF?(z) F(z) 1? g(U).Also, by setting2382Qing-Hua Xu, Ying-Chun Gui and H. M. Srivastavap(z) = 1 +1 ?zF?(z)F(z) 1? ,(15)we deduce thatp(0) = g(0) = 1andp(z) g(U)(z U).Therefore, we have p(z) g(z)(z U). Thus, according to the above Lemma based upon the principle of subordination between analytic functions, we obtain?p(m)(0) m!? |g?(0)|(m N).(16)On the other hand, we find from (15) thatzF?(z) =?1 + p(z) 1?F(z)(z U).(1