Inequalities for $\zeta (s)-\psi (1-s)$ 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 $\zeta (s)-\psi (1-s)$ 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 $\pi$, $\log (2\pi )$ and Stieltjes constants. These results lead to new inequalities for $\zeta (s)-\psi (1-s)$ and shed further light on the conjecture.