| Summary: | By making use of some families of integral and derivative operators, many distinct subclasses of analytic, starlike functions, and symmetric starlike functions have already been defined and investigated from numerous perspectives. In this article, with the help of the one-parameter Bernardi integral operator, we investigate several majorization results for the class of normalized starlike functions, which are associated with the Janowski functions. We also give some particular cases of our main results. Finally, we direct the interested readers to the possibility of examining the fundamental or quantum (or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">q</mi></semantics></math></inline-formula>-) extensions of the findings provided in this work in the concluding section. However, the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="fraktur">p</mi><mo>,</mo><mi mathvariant="fraktur">q</mi><mo>)</mo></mrow></semantics></math></inline-formula>-variations of the suggested <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">q</mi></semantics></math></inline-formula>-results will provide relatively minor and inconsequential developments because the additional (rather forced-in) parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">p</mi></semantics></math></inline-formula> is obviously redundant.
|
|---|