Monotonicity and logarithmic concavity of two functions involving exponential function

Abstract

The function

for x > 0 is proved to be strictly decreasing. As an application of this monotonicity, the logarithmic concavity of the function

for a ∈ ℝ and t ∈ (0, ∞) is verified. The possible origin and background of the function (*) are revealed to be related to

the remainder of Binet's formula. Some applications of above results to the difference of θ(x) are noted.

Keywords:

• monotonicity
• logarithmic concavity
• exponential function
• inequality
• origin
• Binet's formula

2000 Mathematics Subject Classifications:

• 26A48
• 26D07

Acknowledgements

The authors are indebted to the anonymous referees for their many helpful comments on and valuable suggestions to the original version of this manuscript.