Inclusion Properties for Certain Classes of Meromorphic Functions Associated with a Family of Linear Operators
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Research Article Inclusion Properties for Certain Classes of Meromorphic Functions Associated with a Family of Linear Operators Nak Eun Cho Department of Applied Mathematics, Pukyong National University, Pusan 608-737, South Korea Correspondence should be addressed to Nak Eun Cho, [email protected] Received 3 March 2009; Accepted 1 May 2009 Recommended by Ramm Mohapatra The purpose of the present paper is to investigate some inclusion properties of certain classes of meromorphic functions associated with a family of linear operators, which are defined by means of the Hadamard product or convolution. Some invariant properties under convolution are also considered for the classes presented here. The results presented here include several previous known results as their special cases. Copyright q 2009 Nak Eun Cho. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction Let A be the class of analytic functions in the open unit disk U {z ∈ C : |z| < 1} with the usual normalization f0 f 0 − 1 0. If f and g are analytic in U, we say that f is subordinate to g, written f ≺ g or fz ≺ gz, if there exists an analytic function w in U with w0 0 and |wz| < 1 for z ∈ U such that fz gwz. Let N be the class of all functions φ which are analytic and univalent in U and for which φU is convex with φ0 1 and Re{φz} > 0 for z ∈ U. We denote by S ∗ and K the subclasses of A consisting of all analytic functions which are starlike and convex, respectively. Let M denote the class of functions of the form fz
∞ 1 ak zk , z k0
1.1
which are analytic in the punctured open unit disk D U\{0}. For 0 ≤ η, β < 1, we denote by MSη, MKη and MCη, β the subclasses of M consisting of all meromorphic functions which are, respectively, starlike of order η, convex of order η and colse-to-convex of order β and type η in U see, for details, 1, 2.
2
Journal of Inequalities and Applications
Making use of the principle of subordination between analytic functions, we introduce the subclasses MSη, φ, MKη, φ and MCη, β; φ, ψ of the class M for 0 ≤ η, β < 1 and φ, ψ ∈ N, which are defined by zf z 1 − − η ≺ φz in U , 1−η fz
zf z 1 MK η; φ : f ∈ M : − 1 − η ≺ φz in U , 1−η f z MS η; φ :
MC η, β; φ, ψ :
f ∈M:
zf z 1 − − β ≺ ψz in U . f ∈ M : ∃g ∈ MS η; φ s.t. 1−β gz
1.2
We note that the classes mentioned above are the familiar classes which have been used widely on the space of analytic and univalent functions in U see 3–5 and for special choices for the functions φ and ψ involved in these definitions, we can obtain the well-known subclasses of M. For examples, we have 1z MS η , MS η; 1−z 1z MK η; MK η , 1−z 1z 1z , MC η, β; MC η, β . 1−z 1−z
1.3
Now we define the function φa, c; z by φa, c; z :
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