Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
Muhammad Aamir Ali, Hüseyin Budak, Abdullah Akkurt, Yu‐Ming Chu
Abstract
Abstract In this paper, we first prove an identity for twice quantum differentiable functions. Then, by utilizing the convexity of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">∣</m:mo> <m:mrow> <m:mmultiscripts> <m:mrow> <m:mi>D</m:mi> </m:mrow> <m:mrow> <m:mi>q</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mprescripts /> <m:none /> <m:mrow> <m:mi>b</m:mi> </m:mrow> </m:mmultiscripts> <m:mspace width="0.08em" /> <m:mi>f</m:mi> </m:mrow> <m:mo stretchy="false">∣</m:mo> </m:mrow> </m:math> | {}^{b}D_{q}^{2}\hspace{0.08em}f| and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">∣</m:mo> <m:mrow> <m:mmultiscripts> <m:mrow> <m:mi>D</m:mi> </m:mrow> <m:mrow> <m:mi>q</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mprescripts /> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:none /> </m:mmultiscripts> <m:mspace width="0.08em" /> <m:mi>f</m:mi> </m:mrow> <m:mo stretchy="false">∣</m:mo> </m:mrow> </m:math> | {}_{a}D_{q}^{2}\hspace{0.08em}f| , we establish some quantum Ostrowski inequalities for twice quantum differentiable mappings involving <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>q</m:mi> </m:mrow> <m:mrow> <m:mi>a</m:mi> </m:mrow> </m:msub> </m:math> {q}_{a} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>q</m:mi> </m:mrow> <m:mrow> <m:mi>b</m:mi> </m:mrow> </m:msup> </m:math> {q}^{b} -quantum integrals. The results presented here are the generalization of already published ones.