In this work, two enhanced versions of Wirtinger’s inequality are developed. These improvements arise when considering a weighted sum of multiple Wirtinger’s inequalities. Depending on the context, one of the proposed refinements may be applicable than the other. Finally, a simple application of such refinements is presented.
The Hardy-Hilbert integral inequality has inspired a vast body of research over the past few decades, resulting in the creation of numerous new forms and generalizations of integral inequalities. In this article, we build on this line of research by introducing a new class of Hardy-Hilbert-type integral inequalities incorporating an adjustable function. This additional flexibility enables our results to bridge the gap naturally between classical cases and a variety of new ones. We provide several distinct examples to illustrate the applicability and sharpness of the derived inequalities. Additionally, we present a supplementary result that extends the main theorem, supported by concrete examples that demonstrate its validity and scope.
We present a new sharp Ostrowski-type inequality in the L2 norm for functions with absolutely continuous second derivative and third derivative in L2. The inequality depends on two parameters α, γ ∈ [0, 1] and generalizes the sharp inequality of Liu [1]. Special choices of parameters yield known sharp inequalities for midpoint, trapezoid, Simpson, corrected Simpson, and averaged midpoint-trapezoid rules. A complete sharpness proof is given, including explicit verification of the extremal function’s regularity. Applications to composite numerical integration are provided with explicit error bounds, and a numerical example illustrates the theoretical estimates.
This article introduces and analyzes a new class of integral inequalities relating the integrals of two functions over different intervals. Using classical tools such as the Hermite-Hadamard, Steffensen and Young integral inequalities, we derive several refined bounds under monotonicity and convexity assumptions.
This article makes a contribution to the ongoing development of the Steffensen integral inequality by presenting two new results. The first result generalizes the classical Steffensen integral inequality by introducing an additional function that combines key aspects of the Steffensen and Chebyshev integral inequalities. The second result presents a concave integral inequality derived using integration techniques. Numerical examples are provided to demonstrate the validity and application of the results.
