LAUSR

In the paper, the authors aim to present a double inequality for the integral mean $$\begin{aligned} \frac{1}{2\pi }\int _0^{2\pi }a^{\cos ^2\theta }b^{\sin ^2\theta }{{\mathrm{d}}}\theta \end{aligned}$$12π∫02πacos2θbsin2θdθin terms of the exponential and logarithmic means. For attaining the goal, by the Cauchy residue theorem in the theory of complex functions and properties of definite integrals, the authors represent the above integral mean in terms of the modified Bessel function of the first kind. Finally, by virtue of inequalities for the hyperbolic tangent function, the authors further refine upper bounds in the newly-established double inequality in terms of the arithmetic and geometric means.