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For a,b > 0 with a = b , the power-exponential mean is defined by Z ≡ Z (a,b) = exp ( a lna+b lnb a+b ) = √ abe tanht , where t = ln √ a/b . In this paper, we prove the double inequality ( Zp +Gp 2 )1/p < A < ( Zq +Gq 2 )1/q holds for a,b > 0 , a = b with the best constants p = 2/3 and q = 1 , where A = (a+b)/2 , G = √ ab . We also establish the sharp bounds for et tanh t as follows: 1 < et tanht 2cosh t −1 < 1.055, 1 √ 2 < et tanht 2(cosh t) −1 < 1 for t > 0 . These improve some known results. Mathematics subject classification (2010): Primary 26E60, 33B10, Secondary 26D20, 26A48.

Optimal inequalities involving power-exponential mean, arithmetic mean and geometric mean