A new family of multivalent functions defined by certain forms of the quantum integral operator

Abstract In this work, using the concepts of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>q</m:mi> </m:math> q -calculus, we first define the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>q</m:mi> </m:math> q -Jung-Kim-Srivastava and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>q</m:mi> </m:math> q -Bernardi integral operators for multivalent functions. Then, we use these operators to establish the generalized integral operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mrow> <m:mi class="MJX-tex-caligraphic" mathvariant="script">ℬ</m:mi> </m:mrow> <m:mrow> <m:mi>q</m:mi> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mi>m</m:mi> <m:mo>−</m:mo> <m:mi>λ</m:mi> </m:mrow> </m:msubsup> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {{\mathcal{ {\mathcal B} }}}_{q,p}^{-m-\lambda }f\left(z) for multivalent functions. By using the newly defined operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mrow> <m:mi class="MJX-tex-caligraphic" mathvariant="script">ℬ</m:mi> </m:mrow> <m:mrow> <m:mi>q</m:mi> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mi>m</m:mi> <m:mo>−</m:mo> <m:mi>λ</m:mi> </m:mrow> </m:msubsup> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {{\mathcal{ {\mathcal B} }}}_{q,p}^{-m-\lambda }f\left(z) , we define a new class of multivalent analytic functions and diskuss various interesting features of the functions belonging to this class. Certain interesting examples of our developments are also considered. Furthermore, we explore the applications of the Carathéodory lemma in the form of specific results.