Some differential inequalities and Carathéodory functions

AbstractWe consider the volumetric-isochoric split in planar isotropic hyperelasticity and give a precise analysis of rank-one convexity criteria for this case, showing that the Legendre-Hadamard ellipticity condition separates and simplifies in a suitable sense. Starting from the classical two-dimensional criterion by Knowles and Sternberg, we can reduce the conditions for rank-one convexity to a family of one-dimensional coupled differential inequalities. In particular, this allows us to derive a simple rank-one convexity classification for generalized Hadamard energies of the type $W(F)=\frac{\mu }{2} \hspace{0.07em} \frac{\lVert F \rVert ^{2}}{\det F}+f(\det F)$ W ( F ) = μ 2 ∥ F ∥ 2 det F + f ( det F ) ; such an energy is rank-one convex if and only if the function $f$ f is convex.

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The sharp bounds of the second and third Hankel determinants for the class 𝓢𝓛*

Mathematica Slovaca ◽

10.1515/ms-2017-0398 ◽

2020 ◽

Vol 70 (4) ◽

pp. 849-862

Author(s):

Shagun Banga ◽

S. Sivaprasad Kumar

Keyword(s):

Triangle Inequality ◽

Starlike Functions ◽

Sharp Bound ◽

The Novel ◽

Sharp Bounds ◽

Hankel Determinants ◽

Lemniscate Of Bernoulli ◽

Carathéodory Functions ◽

The Right

AbstractIn this paper, we use the novel idea of incorporating the recently derived formula for the fourth coefficient of Carathéodory functions, in place of the routine triangle inequality to achieve the sharp bounds of the Hankel determinants H3(1) and H2(3) for the well known class 𝓢𝓛* of starlike functions associated with the right lemniscate of Bernoulli. Apart from that the sharp bound of the Zalcman functional: $\begin{array}{} |a_3^2-a_5| \end{array}$ for the class 𝓢𝓛* is also estimated. Further, a couple of interesting results of 𝓢𝓛* are also discussed.

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On the critical behavior for inhomogeneous parabolic differential inequalities in the half-space

Applied Mathematics Letters ◽

10.1016/j.aml.2021.107099 ◽

2021 ◽

Vol 117 ◽

pp. 107099

Author(s):

Ravi P. Agarwal ◽

Mohamed Jleli ◽

Bessem Samet

Keyword(s):

Half Space ◽

Critical Behavior ◽

Differential Inequalities

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Nonexistence Results for Higher Order Fractional Differential Inequalities with Nonlinearities Involving Caputo Fractional Derivative

Mathematics ◽

10.3390/math9161866 ◽

2021 ◽

Vol 9 (16) ◽

pp. 1866

Author(s):

Mohamed Jleli ◽

Bessem Samet ◽

Calogero Vetro

Keyword(s):

Fractional Derivative ◽

Caputo Fractional Derivative ◽

A Priori Estimates ◽

Sufficient Conditions ◽

Global Solutions ◽

Higher Order ◽

Differential Inequalities ◽

Structure Of Solutions ◽

Nonlinear Capacity ◽

Fractional Differential

Higher order fractional differential equations are important tools to deal with precise models of materials with hereditary and memory effects. Moreover, fractional differential inequalities are useful to establish the properties of solutions of different problems in biomathematics and flow phenomena. In the present work, we are concerned with the nonexistence of global solutions to a higher order fractional differential inequality with a nonlinearity involving Caputo fractional derivative. Namely, using nonlinear capacity estimates, we obtain sufficient conditions for which we have no global solutions. The a priori estimates of the structure of solutions are obtained by a precise analysis of the integral form of the inequality with appropriate choice of test function.

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Nonexistence of solutions for the quasilinear parabolic differential inequalities with singular potential term and nonlocal source

Journal of Inequalities and Applications ◽

10.1186/s13660-020-02329-5 ◽

2020 ◽

Vol 2020 (1) ◽

Cited By ~ 2

Author(s):

Suping Xiao ◽

Zhong Bo Fang

Keyword(s):

Singular Potential ◽

Differential Inequalities ◽

Nonlocal Source ◽

Potential Term ◽

Nonexistence Of Solutions ◽

Quasilinear Parabolic

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Functional Differential Inequalities with Unbounded Delay

Georgian Mathematical Journal ◽

10.1515/gmj.2005.237 ◽

2005 ◽

Vol 12 (2) ◽

pp. 237-254

Author(s):

Zdzisław Kamont ◽

Adam Nadolski

Keyword(s):

Differential Inequality ◽

Continuous Dependence ◽

Functional Differential Equations ◽

Uniqueness Result ◽

Classical Solutions ◽

Differential Inequalities ◽

Unbounded Delay ◽

Several Variables ◽

Functional Differential ◽

Continuous Dependence Of Solutions

Abstract We prove that a function of several variables satisfying a functional differential inequality with unbounded delay can be estimated by a solution of a suitable initial problem for an ordinary functional differential equation. As a consequence of the comparison theorem we obtain a Perron-type uniqueness result and a result on continuous dependence of solutions on given functions for partial functional differential equations with unbounded delay. We consider classical solutions on the Haar pyramid.

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