The author in [7] conjectured the following inequality; If a and b are nonnegative real numbers with a + b = 1/2, then the inequality 1/2 ≤ a^(2b)k+ b^(2a)k ≤ 1 holds for 0 ≤ k ≤ 1. In this paper, we shall prove the conjecture affirmatively and give the upper and lower estimation of the power exponential functions a^b + b^a for the nonnegative real numbers a and b with a + b = 2. Moreover, we pose some inequalities with power exponential functions.

中文翻译：

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具有幂指数函数的不等式的最佳可能常数

作者在[7]中推测以下不平等。如果一个和b为非负实数与A + B = 1/2，则不等式1/2≤一个^{（图2b）ķ} + B ^{（图2a）ķ} ≤1保持用于0≤ ķ ≤1。在本文中，我们应当肯定地证明猜想和给予的力量指数函数的上部和下部估计一个^b + b^一个用于非负实数一和b与一个+ b =2。此外，我们用幂指数函数带来一些不等式。