Monotonicity of sequences involving geometric means of positive sequences with monotonicity and logarithmical convexity

Let f be a positive function such that x [ f (x + 1)/f (x)− 1 ] is increasing on [1,∞) , then the sequence { n √∏n i=1 f (i) / f (n + 1) }∞ n=1 is decreasing. If f is a logarithmically concave and positive function defined on [1,∞) , then the sequence { n √∏n i=1 f (i) /√ f (n) }∞

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