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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 http://journals.tubitak.gov.tr/math/ Research Article Turk J Math (2013) 37: 437 – 444 ¨ ITAK ˙ c TUB doi:10.3906/mat-1012-543 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: 07.12.2010 • Accepted: 20.04.2012 • Published Online: 26.04.2013 • Printed: 27.05.2013 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 = |fz |2 −|fz |2 = |h (z)|2 −|g (z)|2 . If Jf (z) = |h (z)|2 −|g (z)|2 > 0 , then f = h(z) + g(z) is called a sense-preserving multivalent harmonic function in D. The class of all sense-preserving multivalent harmonic functions with |bm | 0 . If Jf (z) 0 for all z ∈ D, and such that p(z) ∈ P(m) if and only if for some function φ(z) ∈ Ω and every z ∈ D([2], [6]). ∞ Let C(A, B, m) denote the class of functions f(z) = z m + n=m+1 cn z n regular in D and satisfies the condition 1+z f (z) = p(z), f (z) (1.2) for some p(z) ∈ P(A, B, m) and every z ∈ D. Finally, a function f(z) = z m + ∞ n=m+1 dn z n is in the class of K(A, B, m) if there is a function φ(z) in C(A, B, m) such