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Inequalities for $ζ(s)-ψ(1-s)$ related to a conjecture of Henry

In this paper we investigate analytic inequalities related to a conjecture of Henry involving the difference between the Riemann zeta function and the digamma function. By treating $ζ(s)-ψ(1-s)$ as a unified analytic object, we establish its strict convexity and monotonicity on suitable intervals. Moreover, we obtain explicit boundary limits of the derivative, expressed in terms of $π$, $\log (2π)$ and Stieltjes constants. These results lead to new inequalities for $ζ(s)-ψ(1-s)$ and shed further light on the conjecture.