Diophantine Inequality by Unlike Powers of Primes

Suppose that λ1, ⋯ λ5 are nonzero real numbers, not all of the same sign, satisfying that λ1λ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{\lambda _1}} \over {{\lambda _2}}}$$\end{document} is irrational. Then for any given real number η and ε > 0, the inequality |λ1p1+λ2p22+λ3p33+λ4p44+λ5p55+η|<(max1≤j≤5pjj)−19756+ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| {{\lambda _1}{p_1} + {\lambda _2}p_2^2 + {\lambda _3}p_3^3 + {\lambda _4}p_4^4 + {\lambda _5}p_5^5 + \eta } \right| < {\left( {\mathop {\max }\limits_{1 \le j \le 5} p_j^j} \right)^{ - {{19} \over {756}} + \varepsilon }}$$\end{document}