Concavity structure of fractional differential equations and zeros of Mittag–Leffler functions

Abstract In classical calculus, concavity properties of a function are determined by the sign of its second derivative. In this work, we develop a real-variable framework for studying concavity in the nonlocal setting of Riemann–Liouville fractional derivatives of order $$1<\alpha <2$$ . We show that, under homogeneous boundary conditions, the sign of the fractional derivative alone does not determine global concavity. More precisely, (i) if the fractional derivative is nonnegative, non-decreasing, and nontrivial, then the solution is strictly concave up, recovering a classical-type result; (ii) if the fractional derivative is nonnegative and nontrivial, vanishes at the right endpoint, and satisfies a suitable regularity condition near the left endpoint, then the solution is necessarily strictly concave up near the left endpoint and strictly concave down near the right endpoint. In particular, the second derivative vanishes at some point in the interior. These results reveal a fundamental qualitative difference between classical and fractional derivatives. As an application, we establish the existence of a positive real zero of the Mittag–Leffler function $$E_{\alpha ,\alpha -2}(-z)$$ for $$1<\alpha <2$$ . Moreover, denoting by $$\lambda _\beta $$ the first positive real zero of $$E_{\alpha ,\beta }(-z)$$ , we prove the strict ordering $$ \lambda _{\alpha -1}<\lambda _{\alpha -2}<\lambda _\alpha . $$ More broadly, our approach provides a direct method for detecting and comparing real zeros of special functions through geometric properties of solutions to fractional differential equations, offering an alternative to traditional techniques based on complex analysis.