query: Convex inequality with the function $f(x)=x\log x$
positive_passages.docid: 7825
positive_passages.text: If $f$ is a concave function and $f(0) \geq 0$ then $f(a+b) \leq f(a) + f(b)$ for all $a,b >0$
negative_passages.docid: 23119
negative_passages.text: An application of Jensen's Inequality
negative_passages.docid: 116548
negative_passages.text: How to prove the real value function $f$ is convex on $[a,d]$ if $f$ is convex on both $[a,c]$ and $[b,d]$? ($a<b<c<d$).
negative_passages.docid: 77283
negative_passages.text: Cauchy functional inequality
negative_passages.docid: 27570
negative_passages.text: Prove $(1-x)\ln(1-x)+(1+x)\ln(1+x)\leq 2x^2$ for $0<x<1$
negative_passages.docid: 6449
negative_passages.text: Prove logarithmic inequality with greatest integer function.
negative_passages.docid: 85153
negative_passages.text: Show that $f(x) = -\ln(x)$ is convex (WITHOUT using second derivative!)
negative_passages.docid: 55866
negative_passages.text: Convex + Monotone =? Convex
negative_passages.docid: 14448
negative_passages.text: Elementary proof that $|x|^p$ is convex.
negative_passages.docid: 150727
negative_passages.text: Characterization convex function.
negative_passages.docid: 158399
negative_passages.text: Show that $\{(x_1,x_2):x_1x_2\geq4,x_1>0,x_2>0\}$ is a convex set
negative_passages.docid: 151981
negative_passages.text: Tangent line of a convex function
negative_passages.docid: 110146
negative_passages.text: I need to prove the continuity of $f(x)=\log x$ using a $\epsilon-\delta$ proof
negative_passages.docid: 35867
negative_passages.text: Squaring sides of an inequality
negative_passages.docid: 131180
negative_passages.text: When Jensen's inequality is equality
negative_passages.docid: 125175
negative_passages.text: convexity proof of a function including ln and sums
negative_passages.docid: 27587
negative_passages.text: inequality on the unit disk