We present a novel strict inequality for cyclic \(n\)-gons, establishing a relationship between their side lengths, circumradius, and generalized inradius. Specifically, we demonstrate that for any non-degenerate cyclic \(n\)-gon, the inequality \(\left(\frac{a_{1}+a_{2}+\cdots+a_{n-1}}{a_{n}}\right)^{R/r}>\sqrt{e}\) strictly holds. The proof leverages the Padé approximant bound for the natural logarithm and establishes the strict concavity and subadditivity of an associated trigonometric function. We further extend this result to polygons inscribed in ellipses via affine transformations, generalizing the bound in terms of the ellipse’s semi-axes.
We establish scaled block-mean Jensen-type estimates for \(m\)-convex functions in the framework of positive linear functionals. The main observation is that, after a positive functional is decomposed into two nontrivial blocks, the ordinary block identity \(\mu=\alpha\mu_1+\beta\mu_2,\) must be written in the \(m\)-convex geometry as \(\mu=\alpha\mu_1+m\beta\left(\frac{\mu_2}{m}\right).\) Thus the complementary block mean is evaluated at the scaled point \(\mu_2/m\), not at \(\mu_2\). This yields a valid two-block Jensen-type estimate under explicit domain and admissibility assumptions; a simple example shows that the unscaled analogue may fail for \(m<1\). We derive modulus-controlled corrections, approximate stability estimates, superquadratic lower bounds for the scaled block gap, and recursive versions obtained from nested block decompositions. Weighted sums, ordinary integrals, time-scale integrals, Jackson \(q\)-integrals, and positive fractional-kernel operators are obtained as direct realizations of the abstract positive-functional result.
In this work, we derive some new Chebyshev-type integral inequalities involving weighted functions for Hölder-continuous functions. Our results represent a generalization of Chesneau’s. These results will be established by utilizing the properties of bounded functions, monotone functions, and applying Hölder’s inequality with a connection between integral conditions on the derivatives of the functions.
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.
In this paper, integral midpoint type inequalities involving Riemann-Liouville fractional integrals for tgs-convex functions are proved. Two different identities are utilized to get some new integral midpoint type inequalities. One identity is used to obtain\:inequalities for functions whose first derivatives are tgs-convex functions and another identity is used to obtain inequalities for the functions whose second derivatives are tgs-convex. Some numerical examples along with graphical representation are also included to demonstrate the effectiveness of the results. The results demonstrate that the newly established bounds offer significant improvements and tighter estimates compared to existing inequalities in the literature.
We extend the Steffensen-type inequality, proved in the recent paper under concavity and certain smoothness assumptions, to a weighted measure dμ = wdx and a relaxation on smoothness. We also give two numerical examples where the original inequality cannot be applied, while our assumptions are satisfied.
