Horst Alzer and Man Kam Kwong[a]
A log-convexity property of the Riemann zeta function
Pages: 447-457
Received: 24 July 2023
Accepted in revised form: 18 September 2023
Mathematics Subject Classification: 11M06, 26A51, 39B62.
Keywords: Riemann zeta function, log-convex, functional inequality.
Authors address:
[a]: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hunghom, Hong Kong
Abstract: We prove that the function \(x\mapsto \zeta(\cosh(x))\) is strictly log-convex on \((0,\infty).\) An application of this result leads to \[ \zeta\Bigl( \frac{\sqrt{(x+1)(y+1)} + \sqrt{(x-1)(y-1)} }{2} \Bigr) < \sqrt{ \zeta(x) \zeta(y) } \;\ (x,y>1, \, x\neq y) \] which refines the well-known functional inequality \[ \zeta \Bigl( \frac{x+y}{2} \Bigr) < \sqrt{\zeta(x)\zeta(y)} \quad (x,y>1, \, x\neq y). \]