New Post Quantum Analogues of Hermite–Hadamard Type Inequalities for Interval-Valued Convex Functions
Humaira Kalsoom, Muhammad Aamir Ali, Muhammad Idrees, Praveen Agarwal, Muhammad Arif
Abstract
The main objective of this paper is to introduce <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mi>I</a:mi> <a:msub> <a:mrow> <a:mfenced open="(" close=")" separators="|"> <a:mrow> <a:mi>p</a:mi> <a:mo>,</a:mo> <a:mi>q</a:mi> </a:mrow> </a:mfenced> </a:mrow> <a:mrow> <a:mi mathvariant="normal">ϱ</a:mi> </a:mrow> </a:msub> </a:math> -derivative and <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" id="M2"> <g:mi>I</g:mi> <g:msub> <g:mrow> <g:mfenced open="(" close=")" separators="|"> <g:mrow> <g:mi>p</g:mi> <g:mo>,</g:mo> <g:mi>q</g:mi> </g:mrow> </g:mfenced> </g:mrow> <g:mrow> <g:mi mathvariant="normal">ϱ</g:mi> </g:mrow> </g:msub> </g:math> -integral for interval-valued functions and discuss their key properties. Also, we prove the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" id="M3"> <m:mi>I</m:mi> <m:msub> <m:mrow> <m:mfenced open="(" close=")" separators="|"> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mi mathvariant="normal">ϱ</m:mi> </m:mrow> </m:msub> </m:math> -Hermite–Hadamard inequalities for interval-valued functions is the development of <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" id="M4"> <s:msub> <s:mrow> <s:mfenced open="(" close=")" separators="|"> <s:mrow> <s:mi>p</s:mi> <s:mo>,</s:mo> <s:mi>q</s:mi> </s:mrow> </s:mfenced> </s:mrow> <s:mrow> <s:mi mathvariant="normal">ϱ</s:mi> </s:mrow> </s:msub> </s:math> -Hermite–Hadamard inequalities by using new defined <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" id="M5"> <y:mi>I</y:mi> <y:msub> <y:mrow> <y:mfenced open="(" close=")" separators="|"> <y:mrow> <y:mi>p</y:mi> <y:mo>,</y:mo> <y:mi>q</y:mi> </y:mrow> </y:mfenced> </y:mrow> <y:mrow> <y:mi mathvariant="normal">ϱ</y:mi> </y:mrow> </y:msub> </y:math> -integral. Moreover, we prove some results for midpoint- and trapezoidal-type inequalities by using the concept of Pompeiu–Hausdorff distance between the intervals. It is also shown that the results presented in this paper are extensions of some of the results already shown in earlier works. The proposed studies produce variants that would be useful for performing in-depth investigations on fractal theory, optimization, and research problems in different applied fields, such as computer science, quantum mechanics, and quantum physics.