Search

Summary–The simple and elegant result known as Lagrange’s identity can be used to derive some well-known results in mathematics and statistics. We share four such results that have elementary deriv...

Fractional calculus has been the target of the work of many mathematicians for more than a century. Some of these investigations are of inequalities and fractional integral operators. In this article, a novel fractional operator which is known as weighted generalized proportional Hadamard fractional operator with unknown attribute weight is proposed. First, a fractional formulation is constructed, which covers a subjective list of operators. With the aid of the above mentioned operators, numerous notable versions of Polya-Szego, Chebyshev and certain related variants are established. Meanwhile, new outcomes are introduced and new theorems are exhibited. Taking into account the novel generalizations, our consequences have a potential association with the previous results. Furthermore, we demonstrate the applications of new operator with numerous integral inequalities by inducing assumptions on weight function $ \varpi $ and proportionality index $ \varphi. $ It is hoped that this research demonstrates that the suggested technique is efficient, computationally, very user-friendly and accurate.

In this paper, we propose a new framework of weighted generalized proportional fractional integral operator with respect to a monotone function $ \Psi, $ we develop novel modifications of the aforesaid operator. Moreover, contemplating the so-called operator, we determine several notable weighted Chebyshev and Gruss type inequalities with respect to increasing, positive and monotone functions $ \Psi $ by employing traditional and forthright inequalities. Furthermore, we demonstrate the applications of the new operator with numerous integral inequalities by inducing assumptions on $ \omega $ and $ \Psi $ verified the superiority of the suggested scheme in terms of efficiency. Additionally, our consequences have a potential association with the previous results. The computations of the proposed scheme show that the approach is straightforward to apply and computationally very user-friendly and accurate.

This manuscript is concerned with the stability and synchronization for fractional-order delayed gene regulatory networks (FODGRNs) via Razumikhin approach. First of all, the existence of FODGRNs are established by using homeomorphism theory, 2-norm based on the algebraic method and Cauchy Schwartz inequality. The uniqueness of this work among the existing stability results are, the global Mittag-Leffler stability of FODGRNs is explored based on the fractional-order Lyapunov Razumikhin approach. In the meanwhile, two different controllers such as linear feedback and adaptive feedback control, are designed respectively. With the assistance of fractional Razumikhin theorem and our designed controllers, we have established the global Mittag-Leffler synchronization and adaptive synchronization for addressing master-slave systems. Finally, three numerical cases are given to justify the applicability of our stability and synchronization results.

We present the sufficient condition for a classical two-class problem from Fisher discriminant analysis has a solution. Actually, the solution was presented up to our knowledge with a necessary condition only. We use an extended Cauchy–Schwarz inequality as a tool.

Certain subclass of p-valently analytic functions with positive coeffi- cients is introduced, and some interesting properties belonging to the this class is obtained.

We study some interesting characterizations of real inner product spaces expressed in terms of angular distances. We first discuss the equivalence of characterizing an inner product space via the usual angular distance and the p-angular distance. Then, we establish a parametric family of upper bounds for the usual angular distance which also serves as a characterization of an inner product space. As an application, bounds for the usual angular distance are utilized in obtaining improvements of the real Cauchy–Schwarz inequality. Finally, we give several comparative relations for angular distances in inner product spaces.

In this paper we consider two-mode superposition of truncated coherent states. Using this we construct quasi-Bell states and investigate the non-classical features of two-mode superposed truncated coherent states such as oscillatory and sub-Poissonian photon statistics and violation of Cauchy-Schwartz inequality.

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.

In this work, an extension of the generalized mixed Schwarz inequality is proved. A companion of the generalized mixed Schwarz inequality is established by merging both Cartesian and Polar decompositions of operators. Based on that some numerical radius inequalities are proved.

In this paper, we establish 5 kinds of integral Opial-type inequalities in (p, q)-calculus by means of H?lder's inequality, Cauchy inequality, an elementary inequality and some analysis technique. First, we investigated the Opial inequalities in (p, q)-calculus involving one function and its (p, q) derivative. Furthermore, Opial inequalities in (p, q)-calculus involving two functions and two functions with their (p, q) derivatives are given. Our results are (p, q)-generalizations of some known inequalities, such as Opial-type integral inequalities and (p, q)-Wirtinger inequality.

In the paper, we introduce a new partial derivative call it β-partial derivatives as the most natural extensions of the limit definitions of the partial derivative and the β-derivative, which obeys classical properties including: continuity, linearity, product rule, quotient rule, power rule, chain rule and vanishing derivatives for constant functions. As applications, we establish some new Opial-Traple type inequalities for the β-partial derivatives.

Let $$\int _\Omega {\vert {\!\!\;\vert {\,\!A_t h\!\,}\vert \!\!\;}\vert ^2 +\vert {\!\!\;\vert {\,\!B_t^* h\!\,}\vert \!\!\;}\vert ^2}\,d\mu (t)

The symmetric shape of some inequalities between two sequences of real numbers generates inequalities of the same shape in operator theory. In this paper, we study a new refinement of the Cauchy–Bunyakovsky–Schwarz inequality for Euclidean spaces and several inequalities for two bounded linear operators on a Hilbert space, where we mention Bohr’s inequality and Bergström’s inequality for operators. We present an inequality of the Cauchy–Bunyakovsky–Schwarz type for bounded linear operators, by the technique of the monotony of a sequence. We also prove a refinement of the Aczél inequality for bounded linear operators on a Hilbert space. Finally, we present several applications of some identities for Hermitian operators.

The aim of this article is to establish several estimates of the triangle inequality in a normed space over the field of real numbers. We obtain some improvements of the Cauchy&ndash, Schwarz inequality, which is improved by using the Tapia semi-inner-product. Finally, we obtain some new inequalities for the numerical radius and norm inequalities for Hilbert space operators.

First, we show that the non-trivial fuzzy inner product space under the linearity condition does not exist, which means a fuzzy inner product space with linearity produces only a crisp real number for each pair of vectors. If the positive-definiteness is added to the condition, then the Cauchy–Schwartz inequality is also proved.

In this paper, we investigate the $${\mathcal {A}}$$-valued Cauchy–Schwarz inequality and its reverse in a 2-semi inner product $$ {\mathcal {A}}$$-module over a locally $$C^*$$-algebra $$ {\mathcal {A}}$$.

In this note we show that the Wang-Zhang inequality can be naturally applied to obtain an elegant reverse for the classical Schwarz inequality in complex inner product spaces.

We extend the improved Schwarz inequality of Dragomir [1, Theorem 2] to any power p ? 2, ||x||p||y||p-|?x,y?|p ? |det [|?x,e?| |?y,e?|(||y||p - |?x,e?|p)1/p (||y||p- |?y,e?|p)1/p]|p for any vectors x, y, e ? Cn with ||e|| = 1. Applications to n-tuples of complex numbers are also included.

The famous Hardy inequality asserts that if f is a non-negative p-integrable $$(p>1)$$ function on $$(0,\infty )$$ , then $$\begin{aligned} \int _{0}^{\infty }\left( \frac{1}{x}\int _{0}^{x}f(t)dt\right) ^pdx\le \left( \frac{p}{p-1}\right) ^p\int _{0}^{\infty }f(x)^pdx. \end{aligned}$$ We present an external form of the Hardy inequality for Hilbert space operators. Moreover, utilizing the operator log-convex functions, a refinement of the operator Hardy inequality is also given.

SummaryUsing the Cauchy—Schwarz inequality and without using calculus, we give an elementary algebraic derivation of Snell’s law.

We use the Holder inequality for mixed exponents to prove some optimal variants of the generalized Hardy–Littlewood inequality for m-linear forms on \(\ell _{p}\) spaces with mixed exponents. Our results extend recent results of Araujo et al.

The Laub–Ilani inequality [‘A subtle inequality’, Amer. Math. Monthly97 (1990), 65–67] states that $x^{x}+y^{y}\geqslant x^{y}+y^{x}$ for nonnegative real numbers $x,y$. We introduce and prove new trigonometric and algebraic-trigonometric inequalities of Laub–Ilani type and propose some conjectural algebraic-trigonometric inequalities of similar forms.

A Sidon-type inequality is proved for discrete orthonormal systems of special formwhose particular case is theWalsh system.

In this note we show that the Kre¼¬n-Lin triangle inequality can be naturally applied to obtain an elegant reverse for a classical numerical radius power inequality for bounded linear operators on complex Hilbert space due to C. Pearcy.

Let 𝒜 be the class of analytic functions f (z) with the normalized condition f(0) = f 0(0)−1 = 0 in the open unit disk U. Bymaking use of Ruscheweyh derivative operator, a new subclass 𝒜(β1, β2, β3, β4; λ) of f(z) ∈ 𝒜 satisfying the inequality for some complex numbers β1, β2, β3, β4 and for some real λ > 0 is introduced. The object of the present paper is to obtain some properties of the function class 𝒜 (β1, β2, β3, β4; λ). Also the radius problems of satisfies the condition is considered.

The purpose of this paper is to prove certain refinements of Ostrowski&rsquo, s inequality in an inner product space. We study extensions of Ostrowski type inequalities in a 2-inner product space. Finally, some applications which are related to the Chebyshev function and the Gr&uuml, ss inequality are presented.

Theorem. Let u1,u2,…,un and w1,w2,…,wn be real numbers satisfying ui≥wi≥0 for i=1,2,…,n . Then ∑i=1nui2−wi2≤(∑i=1nui)2−(∑i=1nwi)2. Proof. Let a, b be real numbers with a≥b≥0 . We can verify the alg...

In this paper, we introduce the generalized Cauchy-Schwarz inequality for an operator T ∈ L ( H ) T\in {\mathcal {L(H)}} and investigate various properties of operators which satisfy the generalized Cauchy-Schwarz inequality. In particular, every p p -hyponormal operator satisfies this inequality. We also prove that if T ∈ L ( H ) T\in {\mathcal {L(H)}} satisfies the generalized Cauchy-Schwarz inequality, then T T is paranormal. As an application, we show that if both T T and T ∗ T^{\ast } in L ( H ) {\mathcal {L(H)}} satisfy the generalized Cauchy-Schwarz inequality, then T T is normal.

We give an integral version and a refinement of M. Niezgoda’s extension of the variant of Jensen’s inequality given by A. McD. Mercer.

We prove a strong optimal Hardy–Sobolev inequality for the twisted Laplacian on ℂn. The twisted Laplacian is the magnetic Laplacian for a system of n particles in the plane, corresponding to the constant magnetic field. The inequality we obtain is strong optimal in the sense that the weight cannot be improved. We also show that our result extends to a one-parameter family of weighted Sobolev spaces.

We refine some classical inequalities for trigonometric functions, such as Jordan's inequality, Cusa–Huygens's inequality, and Kober's inequality.

In this paper we establish the best constant of an anisotropic Gagliardo–Nirenberg-type inequality related to the Benjamin–Ono–Zakharov–Kuznetsov equation. As an application of our results, we prove the uniform bound of solutions for such a equation in the energy space.

In this paper, by using Jensen’s inequality and Chebyshev integral inequality, some generalizations and new refined Hardy type integral inequalities are obtained. In addition, the corresponding reverse relation are also proved.

We study a certain improved fractional Sobolev–Poincaré inequality on domains, which can be considered as a fractional counterpart of the classical Sobolev–Poincaré inequality. We prove the equivalence of the corresponding weak and strong type inequalities; this leads to a simple proof of a strong type inequality on John domains. We also give necessary conditions for the validity of an improved fractional Sobolev–Poincaré inequality, in particular, we show that a domain of finite measure, satisfying this inequality and a ‘separation property’, is a John domain.

In this paper we establish some vector inequalities for two operators related to Schwarz and Buzano results. We show amongst others that in a Hilbert space H we have the inequality 1 2 [ 〈 | A | 2 + | B | 2 2 x , x 〉 1 / 2 〈 | A | 2 + | B | 2 2 y , y 〉 1 / 2 + | 〈 | A | 2 + | B | 2 2 x , y 〉 | ] ≥ | 〈 Re ( B * A ) x , y 〉 | $${1 \over 2}\left[ {\left\langle {{{\left| {\rm{A}} \right|^2 + \left| {\rm{B}} \right|^2 } \over 2}{\rm{x}},{\rm{x}}} \right\rangle ^{1/2} \left\langle {{{\left| {\rm{A}} \right|^2 + \left| {\rm{B}} \right|^2 } \over 2}{\rm{y}},{\rm{y}}} \right\rangle ^{1/2} + \left| {\left\langle {{{\left| {\rm{A}} \right|^2 + \left| {\rm{B}} \right|^2 } \over {\rm{2}}}} {\rm{x}},{\rm{y}}\right\rangle } \right|} \right] \ge \left| {\left\langle {{\mathop{\rm Re}\nolimits} ({\rm{B}}*{\rm{A}})\,{\rm{x}},{\rm{y}}} \right\rangle } \right|$$ for A, B two bounded linear operators on H such that Re (B*A) is a nonnegative operator and any vectors x, y ∈ H. Applications for norm and numerical radius inequalities are given as well.

We prove a generalization of Jackson's inequality for compact sets in the complex plane admitting both upper and lower bounds for their Green's functions, i.e.?the well known Holder Continuity Property (HCP) and the less known but crucial Łojasiewicz-Siciak inequality (ŁS). Moreover, we show that ŁS is a necessary condition for our Jackson type inequality. The resulting Jackson Property provides us with a large class of sets in the complex plane for which the global and local Markov inequalities are equivalent.

Let M be a complete, simply connected Riemannian manifold with negative curvature. We obtain an interpolation of Hardy inequality and Moser–Trudinger inequality on M. Furthermore, the constant we obtain is sharp.

We present a multidimensional integral inequality related to the Hilbert-type inequality and the Carlson-type inequality. As an application, we obtain a sharper form of the Hilbert-type inequality. Carlson-type inequality is also considered.

In the present paper, the inequality inverse to the Cauchy–Bunyakovskii–Schwarz inequality and generalizing other well-known inversions of this inequality is proved.