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Authors
P.S.Bullen
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Equivalencies of many basic elementary inequalities are given
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We study weighted Poincare and Poincare-Sobolev type inequalities with an explicit analysis on the dependence on the $A_p$ constants of the involved weights. We obtain inequalities of the form $$ left (frac{1}{w(Q)}int_Q|f-f_Q|^{q}wright )^frac{1}{q}le C_well(Q)left (frac{1}{w(Q)}int_Q | abla f|^p wright )^frac{1}{p}, $$ with different quantitative estimates for both the exponent $q$ and the constant $C_w$. We will derive those estimates together with a large variety of related results as a consequence of a general selfimproving property shared by functions satisfying the inequality $$ frac{1}{|Q|}int_Q |f-f_Q| dmu le a(Q), $$ for all cubes $Qsubsetmathbb{R}^n$ and where $a$ is some functional that obeys a specific discrete geometrical summability condition. We introduce a Sobolev-type exponent $p^*_w>p$ associated to the weight $w$ and obtain further improvements involving $L^{p^*_w}$ norms on the left hand side of the inequality above. For the endpoint case of $A_1$ weights we reach the classical critical Sobolev exponent $p^*=frac{pn}{n-p}$ which is the largest possible and provide different type of quantitative estimates for $C_w$. We also show that this best possible estimate cannot hold with an exponent on the $A_1$ constant smaller than $1/p$. We also provide an argument based on extrapolation ideas showing that there is no $(p,p)$, $pgeq1$, Poincare inequality valid for the whole class of $RH_infty$ weights by showing their intimate connection with the failure of Poincare inequalities, $(p,p)$ in the range $0<p<1$.
We revisit weighted Hardy-type inequalities employing an elementary ad hoc approach that yields explicit constants. We also discuss the infinite sequence of power weighted Birman-Hardy-Rellich-type inequalities and derive an operator-valued version thereof.
We prove sharp $ell^q L^p$ decoupling inequalities for arbitrary tuples of quadratic forms. Our argument is based on scale-dependent Brascamp-Lieb inequalities.
This paper deals with some inequalities for trigonometric and hyperbolic functions such as the Jordan inequality and its generalizations. In particular, lower and upper bounds for functions such as (sin x)/x and x/(sinh x) are proved.
Famous Redheffers inequality is generalized to a class of anti-periodic functions. We apply the novel inequality to the generalized trigonometric functions and establish several Redheffer-type inequalities for these functions.
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