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Since the supposed Hermite-Hadamard inequality for a convex function was discussed, its expansions, refinements, and variations, which are called Hermite-Hadamard type inequalities, have been widely explored. The main objective of this article is to acquire new Hermite-Hadamard type inequalities employing the Riemann-Liouville fractional operator for functions whose third derivatives of absolute values are convex and quasi-convex in nature. Some special cases of the newly presented results are discussed as well. As applications, several estimates concerning Bessel functions and special means of real numbers are illustrated.

We obtain some Hölder type trace inequalities for operator weighted geometric mean. Some vector inequalities are also given.

In this paper, we mainly give some quadratic refinements of Young type inequalities. Namely: ( v a + ( 1 − v ) b ) 2 − v ∑ j = 1 N 2 j ( b − a b 2 j − 1 2 j ) 2 ≤ ( a v b 1 − v ) 2 + v 2 ( a − b ) 2 $$\begin{array}{} \displaystyle (va+(1-v)b)^{2}-v{{\sum\limits_{j=1}^N}}2^{j}\Big(b- \sqrt[2^{j}]{ab^{2^{j}-1} }\, \Big)^{2}\leq(a^{v}b^{1-v})^{2}+v^{2}(a-b)^{2} \end{array}$$ for v ∉ [0, 1 2 N + 1 $\begin{array}{} \displaystyle \frac{1}{2^{N+1}} \end{array}$ ], N ∈ ℕ, a, b > 0; and ( v a + ( 1 − v ) b ) 2 − ( 1 − v ) ∑ j = 1 N 2 j ( a − a 2 j − 1 b 2 j ) 2 ≤ ( a v b 1 − v ) 2 + ( 1 − v ) 2 ( a − b ) 2 $$\begin{array}{} \displaystyle (va+(1-v)b)^{2}-(1-v){{\sum\limits_{j=1}^N}}2^{j}\Big(a- \sqrt[2^{j}]{a^{2^{j}-1}b}\, \Big)^{2}\leq(a^{v}b^{1-v})^{2}+(1-v)^{2}(a-b)^{2} \end{array}$$ for v ∉ [1 − 1 2 N + 1 $\begin{array}{} \displaystyle \frac{1}{2^{N+1}} \end{array}$ , 1], N ∈ ℕ, a, b > 0. As an application of these scalars results, we obtain some matrix inequalities for operators and Hilbert-Schmidt norms.

This paper addresses the synchronization issue for the drive-response fractional-order memristor‐based neural networks (FOMNNs) via state feedback control. To achieve the synchronization for considered drive-response FOMNNs, two feedback controllers are introduced. Then, by adopting nonsmooth analysis, fractional Lyapunov’s direct method, Young inequality, and fractional-order differential inclusions, several algebraic sufficient criteria are obtained for guaranteeing the synchronization of the drive-response FOMNNs. Lastly, for illustrating the effectiveness of the obtained theoretical results, an example is given.

In this paper, we establish some new integral inequalities involving general kernels. We obtain the related broad range of fractional integral inequalities. Also, we apply the Young inequality to find new forms of inequalities for generalized kernels. These new and motivated results generalize the results for fractional integrals such that fractional integral of a function with respect to an increasing function, Riemann–Lioville fractional integrals, Erdélyi–Kober fractional integrals, Hadamard fractional integrals, generalized factional integral integrals in addition to the corresponding k-fractional integrals.

Some Choi–Davis–Jensen’s type trace inequalities for convex functions are proved. Also, we generalize these inequalities for any arbitrary operator mean via operator monotone decreasing functions. In particular, we present some new order among $$\mathrm{tr}(\Phi (C)A)$$ and $$\mathrm{tr}(\Phi (C)A^{-1})$$ . New refinements of some power type trace inequalities via reverse and refinement of Young’s inequality are established. Among our results, we obtain new versions of the Holder type trace inequality for any arbitrary operator mean.

In this paper, we present numerous refinements of the Young inequality by the Kantorovich constant. We use these improved inequalities to establish corresponding operator inequalities on a Hilbert space and some new inequalities involving the Hilbert-Schmidt norm of matrices. We also give some refinements of the following Heron type inequality for unitarily invariant norm |||⋅||| and A, B, X ∈ Mn (ℂ): | | | A ν X B 1 − ν + A 1 − ν X B ν 2 | | | ≤ ( 4 r 0 − 1 ) | | | A 1 2 X B 1 2 | | | + 2 ( 1 − 2 r 0 ) | | | ( 1 − α ) A 1 2 X B 1 2 + α ( A X + X B 2 ) | | | , $$\begin{array}{} \begin{split} \displaystyle \Big|\Big|\Big|\frac{A^\nu XB^{1-\nu}+A^{1-\nu}XB^\nu}{2}\Big|\Big|\Big| \leq &(4r_0-1)|||A^{\frac{1}{2}}XB^{\frac{1}{2}}||| \\ &+2(1-2r_0)\Big|\Big|\Big|(1-\alpha)A^{\frac{1}{2}}XB^{\frac{1}{2}} +\alpha\Big(\frac{AX+XB}{2}\Big)\Big|\Big|\Big|, \end{split} \end{array}$$ where 1 4 ≤ ν ≤ 3 4 , α ∈ [ 1 2 , ∞ ) $\begin{array}{} \displaystyle \frac{1}{4}\leq \nu \leq \frac{3}{4}, \alpha \in [\frac{1}{2},\infty ) \end{array}$ and r 0 = min{ν, 1 – ν}.

We establish some new refinements to the Hölder inequality. We then apply them to provide some refinements to the extended Euler’s gamma and beta functions. As another application of our results, we give a new proof of the equivalence between the Hölder inequality and the Cauchy-Schwarz inequality.

This paper deals with the Choi’s inequality for measurable operators affiliated with a given von Neumann algebra. Some Young and Cauchy-Schwarz type inequalities for 𝜏 -measurable operators are also given.

Refining and reversing weighted arithmetic-geometric mean inequalities have been studied in many papers. In this paper, we provide some bounds for the differences between the weighted arithmetic and geometric means, using known inequalities. We improve the results given by Furuichi-Ghaemi-Gharakhanlu and Sababheh-Choi. We also give some bounds on entropies, applying the results in a different approach. We explore certain convex or concave functions, which are symmetric functions on the axis t=1/2.

In this paper, we study the solvability of a very general class of integral equations whose kernel depends on four different functions. Necessary and sufficient conditions for the unique solvability of such integral equations are obtained. To achieve such a goal, the main technique consists in introducing eight new convolutions weighted by multi-dimensional Hermite functions and using them as convolutions somehow associated with our integral equations. In this way, two Young-type inequalities are also obtained.

The theory of inequality in mathematics is an extensive and interesting area of research. The classical Young inequality has a great importance in several fields of mathematics, statistics and other disciplines. Many results were obtained on refinements, generalizations and converses of the Young inequality. In this note, we mainly investigate the improved Young and its reverse inequalities. First, we present a new version of a generalization of reverse Young inequality. In addition, we also get the Young and its reverse inequalities for scalars with the classical Kantorovich constant. Finally, we establish the Young and its reverse inequalities for operators on infinite and finite dimensional Hilbert spaces.

In this article, first, we deduce an equality involving the Atangana–Baleanu (AB)-fractional integral operator. Next, employing this equality, we present some novel generalization of Ostrowski type inequality using the Hölder inequality, the power-mean inequality, Young’s inequality, and the Jensen integral inequality for the convexity of |Υ|. We also deduced some new special cases from the main results. There exists a solid connection between fractional operators and convexity because of their fascinating properties in the mathematical sciences. Scientific inequalities of this nature and, particularly, the methods included have applications in different fields in which symmetry plays a notable role. It is assumed that the results presented in this article will show new directions in the field of fractional calculus.

This paper mainly addresses the containment control issues of linear multi-agent systems with undirected fixed communication topology in the presence of certain perturbations. Based on the relative output information among neighboring agents, the relative state observer are adopted to estimate the the relative state information. By the state estimator and the disturbance estimator, a effective control law is given to make the containment control problem can be solved. Furthermore, with help of algebraic theory, Lyapunov theory and Young Inequality, a sufficient condition is established to guarantee that the multi-agent systems with exogenous disturbance can realize the containment control. Finally, a simulation example is provided to illustrate the efficiency of the theoretical analysis.

The main goal of this article is to present generalized extensions of numerical radius inequalities involving the Euclidean operator radius and the numerical radius of some well-known operator quan...

This article, we explore the asymptotic stability and asymptotic synchronization analysis of fractional order delayed Cohen–Grossberg neural networks with discontinuous neuron activation functions (FCGNNDDs). First, under the framework of Filippov theory and differential inclusion theoretical analysis, the global existence of Filippov solution for FCGNNDDs is studied by means of the given growth condition. Second, by virtue of suitable Lyapunov functional, Young inequality and comparison theorem for fractional order delayed linear system, some global asymptotic stability conditions for such system is derived by limiting discontinuous neuron activations. Third, the global asymptotic synchronization condition for FCGNNDDs is obtained based on the pinning control. At last, two numerical simulations are given to verify the theoretical findings.

Any pair of functions that nearly realizes equality in Young's convolution inequality ∥f∗g∥s≤A∥f∥p∥g∥q with sharp constant, for Euclidean groups, is close in norm to a pair that realizes equality.

We obtain a refined Young type inequality in this paper. The conclusion is presented as follows: Let A, B ∈ B(𝓗) be two positive operators and p ∈ [0, 1], then A ♯ p B + G ∗ ( A ♯ p B ) G ≤ A ∇ p B − 2 r ( A ∇ B − A ♯ B ) , $$\begin{array}{} \displaystyle A\sharp_p B+G^*(A\sharp_p B)G\le A\nabla_p B-2r(A\nabla B-A\sharp B), \end{array}$$ where r = min{p, 1 – p}, G = L ( 2 p ) 2 $\begin{array}{} \displaystyle \frac{\sqrt{L(2p)}}{2} \end{array}$ A –1 S(A|B), L(t) is periodic with period one and L(t) = t 2 2 1 − t t 2 t $\begin{array}{} \displaystyle \frac{t^2}{2}\left( \frac{1-t}{t} \right)^{2t} \end{array}$ for t ∈ [0, 1]. Moreover, we give the s-th powering of two inequalities related to the above one with s > 0 which refines Lin’s work. In the mean time, we present an inequality involving Hilbert-Schmidt norm.

In this paper, we focus on studying the asymptotic stability of the monotone decreasing kink profile solitary wave solutions for the generalized KdV-Burgers equation. We obtain the estimate of the first-order and second-order derivatives for monotone decreasing kink profile solitary wave solutions, and overcome the difficulties caused by high-order nonlinear terms in the generalized KdV-Burgers equation in the estimate by using L2 energy estimating method and Young inequality. We prove that the monotone decreasing kink profile solitary wave solutions are asymptotically stable in H1. Moreover, we obtain the decay rate of the perturbation ψ in the sense of L2 and L∞ norm, respectively, which are (1 + t) −1/2 and (1 + t) −1/4 by using Gargliado-Nirenberg inequality.

This paper proposes a prefiltered observation-based adaptive filter algorithm that is robust against impulsive noise. Previous impulsive noise rejection algorithms were based on output error stochastic, so there was a trade-off relationship between impulsive noise detection and tracking performances. The proposed rejection algorithm is derived by using the statistics of the observed signal and the inequality such as the Schwarz and Young inequality in the absence of impulsive noise. From this, the proposed algorithm updates the weight vector only when the observed signal is not corrupted by impulsive noise. The proposed algorithm achieves the good tracking performance because it distinguishes between the system change and interruption of impulsive noise. In addition, the proposed algorithm has same performance without impulsive noise, compared with the normalized least-mean-square-type algorithm. Further, the proposed rejection algorithm could expand to various adaptive filtering structures, which suffer the performance degradation with impulsive noise, because it is easy to implement. Hence, the proposed algorithm is combined with the NLMS algorithm for dispersive systems and the proportionate NLMS algorithm for sparse systems. Simulation results show that the proposed algorithm achieves fast convergence rate, good tracking performance, and robustness under the impulsive noise environment.

In this article we interpolate Young’s inequality using a delicate treatment of dyadics. Although there are other simple methods to prove these results, we present this new approach hoping to reveal more of the hidden properties of such inequalities.

Using a sharp version of the reverse Young inequality, and a Renyi entropy comparison result due to Fradelizi, Madiman, and Wang (2016), the authors derive Renyi entropy power inequalities for log-concave random vectors when Renyi parameters belong to [0, 1]. Furthermore, the estimates are shown to be sharp up to absolute constants.

In this note, some refinements of Young's inequality and its reverse for positive numbers are proved, and using these inequalities, some operator versions and Hilbert-Schmidt norm versions for matrices of these inequalities are obtained.

In this paper, we present some generalizations of refinements of Young inequalities and reverse inequalities due to Furuichi, Choi as well as Manasrah and Kittaneh to cases of τ-measurable operator...

We present some inequalities for trigonometric sums. Among others, we prove the following refinements of the classical Young inequality. (1) Let m ≥ 3 be an odd integer, then for all n ≥ m − 1 , ∑ k = 1 n cos ⁡ ( k θ ) k ≥ ∑ k = 1 m ( − 1 ) k k . The sign of equality holds if and only if n = m and θ = π . The special case m = 3 is due to Brown and Koumandos (1997). (2) For all even integers n ≥ 2 and real numbers r ∈ ( 0 , 1 ] and θ ∈ [ 0 , π ] we have ∑ k = 1 n cos ⁡ ( k θ ) k r k ≥ − 5 48 ( 5 + 5 ) = − 0.75375 . . . . The sign of equality holds if and only if n = 4 , r = 1 and θ = 4 π / 5 . We apply this result to prove the absolute monotonicity of a function which is defined in terms of the log-function.