New class of normalized multivalent harmonic functions

Let H(m) be the class of functions of the form f(z) = h(z) + <(g(z))over bar>, where f is a harmonic function in a open unit disk U' = {z is an element of C :|z| < 1}. Here, g is the co-analytic part and h is the analytic part of f. The function f is locally univalent in U' if |g'(z)| < |h'(z)| for all z is an element of U', which is the necessary and sufficient condition. A function f belongs to S-H(m) if it is an m-valently harmonic function, where S-H(m) is a subclass of functions f is an element of H(m). Additionally, if Jf =|h`|(2) -|g`|(2) > 0 in U', then the functions in S-H(m) are harmonic and sense-preserving in U'. This paper introduces new classes for normalized multivalent harmonic function within an open unit disk U' = {z is an element of C :|z|<1}, providing a detailed analysis of these functions. The authors investigate properties such as coefficient bounds, which are critical in understanding the behavior of these functions. By normalizing these multivalent harmonic functions, they simplify their form, making it easier to derive and analyze this property. The paper also establishes bounds on the coefficients of the series expansions of these functions, offering significant insights into their structure and limitations. Relevant connections of the results presented here with various known results are briefly indicated.