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Register a new userAnother Proof of a Famous Inequality
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P.S.Bullen
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A simple proof of the weighted two variable geometric-arithmetic a mean inequality based on one given earlier valid only for integer weights
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We show how Turans inequality $P_n(x)^2-P_{n-1}(x)P_{n+1}(x)geq 0$ for Legendre polynomials and related inequalities can be proven by means of a computer procedure. The use of this procedure simplifies the daily work with inequalities. For instance, we have found the stronger inequality $|x|P_n(x)^2-P_{n-1}(x)P_{n+1}(x)geq 0$, $-1leq xleq 1$, effortlessly with the aid of our method.
This paper is intended to give a characterization of the optimality case in Nashs inequality, based on methods of nonlinear analysis for elliptic equations and techniques of the calculus of variations. By embedding the problem into a family of Gagliardo-Nirenberg inequalities, this approach reveals why optimal functions have compact support and also why optimal constants are determined by a simple spectral problem.
We give a direct analytic proof of the classical Boundary Harnack inequality for solutions to linear uniformly elliptic equations in either divergence or non-divergence form.
Let $A$ be a positive semidefinite $mtimes m$ block matrix with each block $n$-square, then the following determinantal inequality for partial traces holds [ (mathrm{tr} A)^{mn} - det(mathrm{tr}_2 A)^n ge bigl| det A - det(mathrm{tr}_1 A)^m bigr|, ] where $mathrm{tr}_1$ and $mathrm{tr}_2$ stand for the first and second partial trace, respectively. This result improves a recent result of Lin [14].
Consider the trilinear form for twisted convolution on $mathbb{R}^{2d}$: begin{equation*} mathcal{T}_t(mathbf{f}):=iint f_1(x)f_2(y)f_3(x+y)e^{itsigma(x,y)}dxdy,end{equation*} where $sigma$ is a symplectic form and $t$ is a real-valued parameter. It is known that in the case $t eq0$ the optimal constant for twisted convolution is the same as that for convolution, though no extremizers exist. Expanding about the manifold of triples of maximizers and $t=0$ we prove a sharpened inequality for twisted convolution with an arbitrary antisymmetric form in place of $sigma$.
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