Quantum Integral Inequalities with Respect to Raina’s Function via Coordinated Generalized<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"><a:mi>Ψ</a:mi></a:math>-Convex Functions with Applications
Saima Rashid, Saad Ihsan Butt, Shazia Kanwal, Hijaz Ahmad, Miao-Kun Wang
Abstract
In accordance with the quantum calculus, we introduced the two variable forms of Hermite-Hadamard- ( <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M2"><a:mi mathvariant="script">H</a:mi><a:mi mathvariant="script">H</a:mi></a:math> -) type inequality over finite rectangles for generalized <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" id="M3"><e:mi>Ψ</e:mi></e:math> -convex functions. This novel framework is the convolution of quantum calculus, convexity, and special functions. Taking into account the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" id="M4"><g:msub><g:mrow><g:mover accent="true"><g:mi>q</g:mi><g:mo stretchy="true">^</g:mo></g:mover></g:mrow><g:mrow><g:mn>1</g:mn></g:mrow></g:msub><g:msub><g:mrow><g:mover accent="true"><g:mi>q</g:mi><g:mo stretchy="true">^</g:mo></g:mover></g:mrow><g:mrow><g:mn>2</g:mn></g:mrow></g:msub></g:math> -integral identity, we demonstrate the novel generalizations of the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" id="M5"><m:mi mathvariant="script">H</m:mi><m:mi mathvariant="script">H</m:mi></m:math> -type inequality for <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" id="M6"><q:msub><q:mrow><q:mover accent="true"><q:mi>q</q:mi><q:mo stretchy="true">^</q:mo></q:mover></q:mrow><q:mrow><q:mn>1</q:mn></q:mrow></q:msub><q:msub><q:mrow><q:mover accent="true"><q:mi>q</q:mi><q:mo stretchy="true">^</q:mo></q:mover></q:mrow><q:mrow><q:mn>2</q:mn></q:mrow></q:msub></q:math> -differentiable function by acquainting Raina’s functions. Additionally, we present a different approach that can be used to characterize <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" id="M7"><w:mi mathvariant="script">H</w:mi><w:mi mathvariant="script">H</w:mi></w:math> -type variants with respect to Raina’s function of coordinated generalized <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" id="M8"><ab:mi>Ψ</ab:mi></ab:math> -convex functions within the quantum techniques. This new study has the ability to generate certain novel bounds and some well-known consequences in the relative literature. As application viewpoint, the proposed study for changing parametric values associated with Raina’s functions exhibits interesting results in order to show the applicability and supremacy of the obtained results. It is expected that this method which is very useful, accurate, and versatile will open a new venue for the real-world phenomena of special relativity and quantum theory.