In this study, we give a new \(m\)-convex function that is called an \(m\)-convex of the second type and its some properties. Moreover, some integral inequalities are examined for each \(m\)-convex function of the second type.

We study finite weighted versions of Cauchy-type inequalities and their relation to the Chebyshev functional. The main elementary device is the reflection of a weighted sequence with respect to its weighted mean. This reflection preserves the total weighted mean and the weighted second mixed moment. We apply the resulting estimates to functions sampled at Fibonacci nodes and to several Fibonacci and Lucas choices of weights and moments.

This paper contributes to the study of weighted semi-norms and their role in integral inequalities, continuing our earlier investigations based on convexity techniques. By incorporating Sonin’s identity into a weighted semi-norm setting, we obtain a unified extension of several classical inequalities of Čebyšev type. The proposed framework allows us to generalize and refine a number of well-known results, including inequalities associated with Čebyšev, Grüss, Ostrowski, and Lupaş, while placing previous contributions by Dragomir and others in a broader weighted context. In particular, we establish new bounds for the weighted Čebyšev functional \(T_w(f,g)\) expressed in terms of the semi-norm \(\Delta_p(f)\), which captures the global oscillatory behavior of the underlying function. Additional improvements are obtained through the use of a weighted Hölder–Işcan type inequality. The resulting theory not only encompasses the classical, unweighted case as a special situation but also offers greater adaptability in problems involving non-uniform weights, probabilistic measures, and weighted approximation processes. As an illustration of applicability, several consequences for numerical integration are discussed, including generalized midpoint and trapezoidal bounds.

Studies on inner-product-type integral transformers have been considered in many research works from various perspectives, including spectra, numerical ranges, and operator inequalities. An open problem remains concerning inequalities related to norm estimates for inner-product-type integral transformers whose spectra are contained in the unit disc. It has been observed that the norms of such transformers can be attained under the condition that one of the implementing operators is normal. In this note, we address this problem by establishing norm inequalities for inner-product-type integral transformers in a general Banach space setting.

This study aims to extend the classical Hermite-Hadamard-type inequalities by employing recently introduced \((k,l)\)-type fractional integrals, which are formulated within the framework of the Riemann-Liouville approach. These integrals are characterized by two exponential parameters, \(k\) and \(l\), defined via the \((k,l)\)-gamma function. In particular, we established new inequalities involving the arithmetic, geometric, and harmonic \((k,l)\)-Riemann-Liouville fractional integrals. Notably, when \(k=l\), these integrals reduce to \(k\)-Riemann-Liouville fractional integrals. Additionally, several foundational identities related to the general \((k,l)\)-Riemann-Liouville fractional integrals are presented. Subsequently, various related inequalities are established using the convexity properties of differentiable functions. These results contribute to the field of fractional calculus and its role in mathematical analysis.