March 3, 2026Open Access

On a class of harmonic functions associated with the µth order differential subordination

Puntos clave

Harmonic functions satisfying the defined coefficient condition establish necessary and sufficient criteria for inclusion in the subclass.
A detailed examination of bounds and extreme points informs our understanding of these harmonic functions' properties.
The formulation involves convolution and integral operators, focusing on the intricacies of the subclass NµPμ(α, β).
Findings reveal significant inequalities for hypergeometric harmonic functions, particularly approaching the case where µ equals 4.

Resumen

In this paper, a class P_^ (, ) of functions f = h + g which are the harmonic shears of the analytic functions h + g is defined and studied. A sufficient coefficient condition for functions f = h + g P_^ (, ) is obtained. It is proved that this coefficient condition is necessary and sufficient for functions belonging to its subclass NP_^ (, ). Results on bounds, extreme points, convolution, convex combinations and integral operator for functions belonging to the subclass NP_^ (, ) are obtained. Inequalities for certain hypergeometric harmonic functions belonging to these classes are also given and the results based on one special case when = 4 are included.

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