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MathCraft: PDF to LaTeX file: Sample-01

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\large\textbf{1. Introduction and Preliminaries}}\end{center} \vspace{2mm} Let us consider the followin... ^{r}-y^{r}\right)} \\ & E(x, y ; 0,0)=\sqrt{x y} \end{aligned} $$ where $x$ and $y$ are positive real ... n*} E(x, y ; r, s) \leq E(x, y ; u, v) . \tag{1} \end{equation*} \vspace{2mm} In this paper, first we ... 2. A Simple Proof of Stolarsky Means Inequality}}\end{center} \vspace{2mm} Note that $E(r, s)$ is cont

MathCraft: PDF to LaTeX file: Sample-02

5 Hits, Last modified: 3 months ago

ig01} %\caption{Caption 1} \label{fig:wrapfig} \end{wrapfigure} \vspace{.2 in} Let now $r(x)=f\left... b} f(x) d x \leq \int_{a}^{b} s(x) d x . \tag{1} \end{equation*} Now $$ \begin{aligned} \int_{a}^{b} r... ght) d x \\ & =f\left(\frac{a+b}{2}\right)(b-a), \end{aligned} $$ and $$ \begin{aligned} \int_{a}^{b} s... a}^{b}(x-a) d x \\ & =\frac{f(a)+f(b)}{2}(b-a) . \end{aligned} $$ Using above value in (1), we have $$