TAILIEUCHUNG - Growth and distortion theorems for multivalent Janowski close-to-convex harmonic functions with shear construction method

In this paper we introduce the class of m-valent Janowski close to convex harmonic functions. Growth and distortion theorems are obtained for this class. Our study is based on the harmonic shear methods for harmonic functions. | Turkish Journal of Mathematics Research Article Turk J Math (2013) 37: 437 – 444 ¨ ITAK ˙ c TUB doi: Growth and distortion theorems for multivalent Janowski close-to-convex harmonic functions with shear construction method ∗ ˘ ¨ Ya¸sar POLATOGLU, Hatice Esra OZKAN, Emel YAVUZ DUMAN ˙ ˙ Department of Mathematics and Computer Science, Istanbul K¨ ult¨ ur University, Istanbul, Turkey Received: • Accepted: • Published Online: • Printed: Abstract: In this paper we introduce the class of m -valent Janowski close to convex harmonic functions. Growth and distortion theorems are obtained for this class. Our study is based on the harmonic shear methods for harmonic functions. Key words: Multivalent harmonic functions, distortion theorem, growth theorem 1. Introduction Let U be a simply connected domain in the complex plane. A harmonic function f has the representation f = h(z) + g(z), where h(z) and g(z) are analytic in U and are called the analytic and co-analytic part of f , respectively. Let h(z) = z m +am+1 z m+1 +am+2 z m+2 +· · · , and g(z) = bm z m +bm+1 z m+1 +bm+2 z m+2 +· · · be analytic functions in the open unit disc D. The jacobian Jf of f = h(z) + g(z) is defined by Jf =

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