时间尺度上反向Callebaut、Rogers Hölder和Cauchy Schwarz不等式的核型分数扩展
Kernel-Type Fractional Extensions of Reverse Callebaut, Rogers Hölder and Cauchy Schwarz Inequalities on Time Scales
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中文总结 AI 辅助
研究时间尺度上反向Callebaut等不等式的核型分数扩展，借助与钻石-α积分相关的非负核算子，能统一处理连续、离散、量子和分数动态环境下的反向不等式，还可用于相关方程的先验估计。
中文摘要 AI 辅助
本文建立了时间尺度上反向Callebaut、Rogers Hölder和Cauchy Schwarz不等式的核型分数扩展。通过与钻石-α积分相关的非负核算子给出结果，能在动态不等式框架内纳入记忆效应。选择特定核和时间尺度，所得不等式可化为连续、离散和量子情形。核为恒等核时可恢复相应经典钻石-α动态不等式。还导出了一些特殊情形结果，可用于研究多种动态环境下的反向不等式及获取相关方程的先验估计。
英文摘要
In this paper, we establish kernel-type fractional extensions of reverse Callebaut, Rogers--Hölder and Cauchy--Schwarz inequalities on time scales. The proposed results are formulated by means of a nonnegative kernel operator associated with the diamond-$α$ integral, which enables the inclusion of memory effects within the framework of dynamic inequalities. By choosing particular kernels and time scales, the obtained inequalities reduce to their continuous, discrete and quantum counterparts. Moreover, when the kernel is taken as the identity kernel, the results recover the corresponding classical diamond-$α$ dynamic inequalities. Several consequences are derived as special cases, including fractional reverse Cauchy--Schwarz inequalities and weighted reverse Rogers--Hölder inequalities. The results provide a unified approach for studying reverse inequalities in continuous, discrete, quantum and fractional dynamic settings and may be applied to obtain a priori estimates for Volterra-type and delay dynamic equations.
