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### Estimate of a ratio of two incomplete gamma functions

I would like to bound from above the expression $$\frac{\Gamma(\alpha,x)-\Gamma(\alpha,y)}{\Gamma(\beta,x)-\Gamma(\beta,y)}$$ for $x>y>0$. By plotting the above expression I have found that ...
I came across these inequalities while learning about Schwartz functions (Classical Fourier Analysis, Grafakos) and I have no idea how to prove this: For $x \in \mathbb{R}^{n}$ and $\alpha = ... 5answers 764 views ### Understanding Gibbs's inequality Short version Gibbs's inequality is a simple inequality for real numbers, usually understood information-theoretically. In the jargon, it states that for two probability measures on a finite set, ... 0answers 300 views ### A curious inequality Let$r_k>0$for$k = 1,\ldots, n$, let$\alpha_k, \beta_k\in \mathbb{R}$be given such that$|\alpha_k|\le \beta_k\le \frac{\pi}{2}$. Suppose further that ... 1answer 205 views ### Inequality with even powers of trigonometric functions For$m>0$,$0 < n\leqslant m+1$($m,n\in \mathbb{Z} $) , and$0 < a < 1$, prove that $$2^{n}\cdot \left( a^{n}\cos ^{2m}\dfrac {\pi a} {2}+\left( 1-a\right) ^{n}\sin ^{2m}\dfrac {\pi ... 3answers 290 views ### Asymptotic behaviour/upper bound for \int_0^{\infty} \exp(-c x^a+K x^b)dx for a>b>0 as K\rightarrow \infty? What is the asymptotic behaviour or an upper bound for \int_0^{\infty} \exp(-c x^a+K x^b) \, dx, for a>b>0, as K\rightarrow \infty? Or any good reference for tools to tackle this question? ... 2answers 262 views ### how to solve a singular integral equation involving the kernel 1/x Dear all, Suppose we know that f(x) is nonnegative and HÃ¶lder continuous at zero with exponents 1/2. We also know that$$ f(x) \le g(x) + \int_0^x \frac{f(y)}{y} d y,\quad\forall x>0, $$... 1answer 146 views ### Strengthening an inequality Let k be an integer. The following inequality is standard.$$ (a+b)^{k+1} - b^{k+1} \leq (k+1)a(a+b)^k $$for a,b > 0. However, does the following inequality still hold$$ (a+b)^{k+1} - ... 1answer 297 views ### sobolev embedding theorem in the smooth metric measure space we know the sobolev embedding theorem of Saloff-Coste$\Big(\int_B|F|^{2q}d\mu\Big)^{\frac1q}\le e^{C(1+\sqrt KR)}V^{-2/n}R^2\int_B\Big(|\nabla F|^2+R^{-2}F^2\Big)d\mu $wtih$Ric\ge-(n-1)K$, for ... 2answers 219 views ### Bounding the series of the geometric means of the terms of a given positive series Let$ \{ a _ k \} _{k\in\mathbb{N} _ +} $be a sequence of non-negative numbers, and let$MG(a_1,\dots,a_n)$denote the geometric mean of the first$n$terms. Then, the inequality $$\sum _ {n\ge ... 3answers 878 views ### Poincare Metric on Hyperbolic Plane as is well known, we can put a metric on the upper half plane \mathbb{R}^+ \times \mathbb{R} by setting$$ d\left((x,t);(x',t')\right):=\log\left(\frac{1 + \delta}{1 - \delta}\right)^{1/2}, $$... 1answer 413 views ### What would the best treatment of Gehring's lemma look like? In a course about elliptic regularity probably one sooner or later stubles into the reverse Holder inequalities, and has to introduce the Gehring lemma, which in one of its many versions improves a ... 1answer 188 views ### An inequality for a continuous non-smooth function Hello, I have a question about how to prove a lemma such as this one, For any 0<\alpha<1 and M_{0}>0, there exists a M_{1}>0 such that \left|z\right|^{\alpha}\leq ... 0answers 67 views ### Bound of polynomial on product space in terms of values on the diagonal We work in the multivariate case so that x stands for (x_1,\ldots, x_n). Let q(x,y) be a symmetric matrix representation of a homogeneous polynomial f(x) of degree d. Explicitly, (q is ... 5answers 921 views ### An inequality on concave functions Could somebody help me to answer the following question? Let f:R_+ \rightarrow R_+ be a nonindentically zero, nondecreasing, continuous, concave function with f(0)=0. Do we have that for any ... 1answer 342 views ### Is there a complex analog of this sharpened Cauchy Inequality? Let x and y be two points on the unit sphere S^{n-1} in Euclidean space {\mathbb{R}}^n. Suppose that the angle \theta between the points x and y is acute, so that the dot product x\cdot ... 1answer 390 views ### Coefficient bounds of an inequality Hello, Given positive integers k and n. Are there upper bounds on coefficients A and B such that they depends only on k (eg., 2 k^k) and for all non-negative integer sequences ... 2answers 1k views ### Question on eigenvalue square root subadditivity ORIGINAL QUESTION Let \lambda_{1}\left(\cdot\right) be the larger eigenvalue of a 2\times2 matrix and \lambda_{2}\left(\cdot\right) the smaller eigenvalue of a 2\times2 matrix. Is it true ... 0answers 443 views ### Any similar inequality in literature? I got the following inequality: B_{4} is the unit ball in R^{4}, \partial B_{4} is its boundary. (\int_{B_{4}}e^{4u}dx)^{\frac{1}{4}} \leq S(\int_{\partial B_{4}}e^{3u}d\xi)^{\frac{1}{3}}, ... 1answer 1k views ### How to estimate derivatives of multivariate polynomial near a manifold I have to provide a (Markov or Bernstein-based?) inequality that gives an upper bound for the partial derivatives of a multivariate polynomial calculated near a real smooth surface in terms of value ... 5answers 520 views ### Binary operations compatible with the usual order on the reals An officemate passes along the following natural-seeming question: Say that a binary operation \oplus is compatible with the usual order \leq on \mathbb{R} if for any w, x, y, z in ... 6answers 3k views ### Applications of Hardy's inequality Every so often I would encounter Hardy's inequality: Theorem 1 (Hardy's inequality). If p>1, a_n \geq 0, and A_n=a_1+a_2+\cdots+a_n, then$$\sum_{n=1}^\infty ... 1answer 322 views ### Statistical inequality Let$X$be a finite discrete variable and$X\ge0$. Is it true that $$16\operatorname{Var}(X) \le \left[8{\mathbb E}(X) + \operatorname{Range}(X)\right]\operatorname{Range}(X)$$ where ... 2answers 584 views ### Logarithm of AM/GM ratio:$\sqrt{\log((x+y)/(2\sqrt{xy}))}$Recently, while playing around with infinite-divisibility, i arrived at the following metric: $$d(x,y) := \sqrt{\log\left(\frac{x+y}{2\sqrt{xy}}\right)},$$ defined for positive reals$x$and$y$. ... 0answers 934 views ### Inequality concerning absolute value of a polynomial Let $$f(z) = (1-1/t) z^w + z/t - 1$$ with integers$t\geq2$and$w\geq2$.Let$r=1+1/(tw^3)$. How do I show $$\left\lvert f(r e^{i\varphi}) \right\rvert \geq \left\lvert f(r) \right\rvert$$ for any ... 1answer 264 views ### Denominators in the solution to Hilbert's XVII Hilbert's seventeenth problem asks to prove that every positive semidefinite form can be written as the sum of squares of rational functions. Currently we don't seem to have a good understanding of ... 2answers 749 views ### Positivity of sequences via generating series There are different ways of showing that a given sequence$a_0,a_1,a_2,\dots$of integers, say, is nonnegative. For example, one can show that$a_n$count something, or express$a_n$as a (multiple) ... 1answer 325 views ### A plausible positivity After getting stuck with the previous positivity (it probably sounds too complex), I would like to give a version of the problem which is of most interest to me. Consider a sequence of real numbers ... 1answer 305 views ### Positivity of “harmonic” summation The settings for the problem are as follows. Given a real number$\alpha\in[0,1]$, consider a sequence of real (positive, negative and zero) numbers$a_1,a_2,\dots,a_n,\dots$satisfying (1)$a_1=1$, ... 3answers 355 views ### monotonicity from 4 term-recursion. In determining the monotonicity of coefficients in a series expansion (which appeared in one of my study), I come across the following problem. Let$p\ge 2$be an integer, and ... 3answers 543 views ### l^p space inequality related to compressed sensing I'm trying to read Donoho's 2004 paper Compressed Sensing and am having trouble with a supposedly trivial statement (equation 1.2 on page 3). He makes the sparsity assumption on$\theta \in ...
Suppose we have a polynomial function $f(z)=a_0+a_1z+a_2z^2+...+z^n$ with each $a_i$ between 0 and 1. Is there a method to determine if $f$ has a pair of complex conjugate roots? There are many ...