Abstract
Let be a positive integer and , , , and real numbers satisfying and . It is proved that for the real numbers the maximum of the function is attained if and only if of the numbers are equal to and the other are equal to , while is one of the values , , where denotes the integer part and represents the Stolarsky mean of and of powers and Some asymptotic results concerning are also discussed.
Highlights
Given the positive real numbers a and b and the real numbers p and q, the difference mean or Stolarsky mean Dp,q a, b of a and b is defined by see, e.g., 1 or 2 Dp,q a, b :
In the following theorem we examine the behavior of k n when the numbers p, q in Theorem 2.1, are terms of a sequence with certain properties
We find the configuration where the maximum is realized
Summary
Let n be a positive integer and p, q, a, and b real numbers satisfying p > q > 0 and 0 < a < b. An ap1 · · · apn /n − aq1 · · · aqn /n p/q is attained if and only if k n of the numbers a1, . The result enables one to obtain elegant proofs for some related inequalities.
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