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The method of exponential sums is one of a few general methods enabling us to solve a wide range of miscellaneous problems from the theory of numbers and its applications. The strongest results have been obtained with the aid of this method. Therefore knowledge of the fundamentals of theory of exponential sums is necessary for studying modern number theory.

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if $a , b$ are positive numbers such that $a + b = 1$ prove that for all $x$ $ae^{\frac{x}{a... [closed]

I am working to prove the following inequality, however I have not reached to any definite solutions. I would be so thankful if anyone could help me prove this inequality or at least give me some hint …