Kernel-Type Fractional Extensions of Reverse Callebaut, Rogers Hölder and Cauchy Schwarz Inequalities on Time Scales

Technical Summary: Kernel-Type Fractional Extensions of Reverse Inequalities on Time Scales

Problem Statement
The paper addresses a gap in the theory of dynamic inequalities on time scales. While reverse forms of classical inequalities (specifically Callebaut, Rogers–Hölder, and Cauchy–Schwarz) have been established within the diamond-$\alpha$ calculus framework, existing results generally lack the incorporation of fractional memory effects or general kernel operators. Many physical and engineering models involving hereditary behavior, nonlocal interactions, and delay require integral operators with kernels to describe the dependence of the current state on past history. The authors aim to extend reverse dynamic inequalities to a unified framework that includes these memory-dependent and kernel-weighted effects, thereby bridging continuous, discrete, and quantum dynamic settings with fractional calculus.

Methodology
The authors employ the theory of time scales ($\mathbb{T}$), which unifies continuous ($\mathbb{T}=\mathbb{R}$) and discrete ($\mathbb{T}=\mathbb{Z}$) analysis, along with the diamond-$\alpha$ calculus. The diamond-$\alpha$ integral is defined as a convex combination of the delta ($\Delta$) and nabla ($\nabla$) integrals, parameterized by $\alpha \in [0, 1]$.

The core methodological innovation is the introduction of a nonnegative kernel-type diamond-$\alpha$ integral operator, defined as:
$$I^\alpha_K h(t) = \int_a^t K(t, s)h(s) \diamond_\alpha s$$
where $K(t, s) \geq 0$ is a suitable kernel function.

The derivation of the inequalities relies on the following steps:

1. Reverse Young Inequalities: The authors utilize refined reverse Young inequalities (specifically those by Kittaneh and Manasrah) and bounded/Kantorovich-type variants. These provide pointwise estimates involving terms like $(1-\delta)\Phi + \delta\Psi - \Phi^{1-\delta}\Psi^\delta$.
2. Kernel Integration: By assuming the kernel $K(t, s)$ is nonnegative, the authors preserve the order of inequalities when integrating. They apply the reverse Young estimates to specific functions of $f$ and $g$ (e.g., $|f(s)|^2|g(r)|^2$), multiply by the product of kernel-weighted terms $\Omega_t(s)\Omega_t(r)$, and integrate over the time scale using the diamond-$\alpha$ integral.
3. Limiting Cases: The framework is designed such that specific choices of the kernel (e.g., $K(t, s) = 1$) and the time scale ($\mathbb{R}, \mathbb{Z}, q^{\mathbb{N}_0}$) reduce the general results to known classical, discrete, and quantum forms.

Key Contributions and Results
The paper establishes several new inequalities that generalize existing reverse dynamic inequalities:

1. Kernel-Type Reverse Callebaut Inequality:
   Theorem 3.1 establishes a double-sided bound for the difference between the product of weighted integrals and the square of a mixed integral. The bounds are determined by constants $\beta = \min\{1-\delta, \delta\}$ and $\gamma = \max\{1-\delta, \delta\}$.

   • Bounded Version: Theorem 3.4 provides an upper bound for the reverse Callebaut gap when the ratio $|f(s)|/|g(s)|$ is bounded between constants $m$ and $M$.

2. Kernel-Type Reverse Rogers–Hölder Inequality:
   Theorem 3.7 derives a reverse Rogers–Hölder inequality involving conjugate indices $p$ and $q$. It bounds the difference between the product of $L^p$ and $L^q$ norms and the mixed integral term.

   • Bounded Version: Theorem 3.10 provides a multiplicative bound involving the Kantorovich ratio when the functions $f$ and $g$ are bounded.
   • Kantorovich-Type: Theorem 3.13 presents a multiplicative reverse form using the Kantorovich constant $K_\gamma(L)$, dependent on the bounds of the functions.

3. Kernel-Type Reverse Cauchy–Schwarz Inequality:
   As a direct consequence of the Rogers–Hölder results (by setting $p=q=2$), the paper derives reverse Cauchy–Schwarz inequalities (Corollary 3.6 and Corollary 3.12). These quantify the deviation from the equality case in the presence of kernel weights.

4. Validation:
   Section 4 validates the theoretical results through explicit calculations on two specific time scales:

   • Continuous Case ($\mathbb{T}=\mathbb{R}$): The diamond-$\alpha$ integral reduces to the standard Riemann integral. Numerical verification confirms the inequality holds for specific polynomial functions and a linear kernel.
   • Discrete Case ($\mathbb{T}=\mathbb{Z}$): The integral reduces to a finite sum. Numerical verification confirms the inequality holds for discrete sequences.

Significance and Claims
The authors claim that the primary significance of this work lies in providing a unified approach for studying reverse inequalities across continuous, discrete, quantum, and fractional dynamic settings.

• Unification: By utilizing the time scale calculus and the diamond-$\alpha$ integral, the results recover classical continuous, discrete, and quantum versions as particular cases simply by choosing the appropriate time scale and kernel.
• Memory Effects: The introduction of the nonnegative kernel operator allows the framework to model memory and hereditary behaviors, which are absent in standard dynamic inequalities.
• Applications: The paper suggests that these inequalities can be used to derive a priori estimates for Volterra-type and delay dynamic equations on time scales. The authors note that the results extend the scope of reverse inequalities from the classical diamond-$\alpha$ setting to a broader kernel-type fractional framework.

The paper concludes by identifying future directions, such as extending these results to fractional operators involving Caputo, Hilfer, or conformable derivatives, and applying the inequalities to stability analysis for nonlinear fractional dynamic equations with delay and impulsive effects. However, the current work focuses strictly on the establishment and validation of the kernel-type reverse inequalities themselves.