Abstract
Complex-valued harmonic functions that are univalent and sense preserving in the open unit disk can be written in the form $f=h+\overline{g}$ , where h and g are analytic. In this paper we investigate some classes of univalent harmonic functions with varying coefficients related to Janowski functions. By using the extreme points theory we obtain necessary and sufficient convolution conditions, coefficients estimates, distortion theorems, and integral mean inequalities for these classes of functions. The radii of starlikeness and convexity for these classes are also determined.
Highlights
1 Introduction Harmonic functions are famous for their use in the study of minimal surfaces and play important roles in a variety of problems in applied mathematics
Clunie and Sheil-Small [ ] pointed out that a necessary and sufficient condition for f to be locally univalent and sense preserving in D is that |h (z)| > |g (z)| in D
Note that for f = h + g, harmonic and sense preserving in the open unit disk D = {z ∈ C : |z| < }, the condition h ( ) = > |g ( )| implies that the function (f – g ( )f )/( – |g ( )| ) is harmonic and sense preserving in D
Summary
Harmonic functions are famous for their use in the study of minimal surfaces and play important roles in a variety of problems in applied mathematics (e.g. see Choquet [ ], Dorff [ ], Duren [ ] or Lewy [ ]). We let H be the class of functions f = h + g, harmonic, sense preserving, and univalent in the open unit disk D, for which fz( ) = g ( ) = . It is the aim of this paper to obtain necessary and sufficient convolution conditions, coefficient bounds, distortion theorems, radii of starlikeness and convexity, compactness, and extreme points for the above defined classes Hλ(A, B) and Gλ(A, B).
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