Application of the q-derivative operator to a specialized class of harmonic functions exhibiting positive real part

This paper introduces a new subclass of harmonic functions with a positive real part, denoted by $H{P}_q(\beta )$, where $0\le \beta <1$ and $0<q<1$. A sufficient coefficient condition is established for functions within this class, which is also necessary when dealing with negative coefficients. In addition, the growth theorem is derived, and the extreme points associated with this subclass are also identified. Finally, the $q$-integral operator for harmonic functions of the form $f=h+g$ with a positive real part is presented.