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In this paper, we consider the backward problem for fractional in timeevolution equations $\partial_t^\alpha u(t)= A u(t)$ with the Caputo derivativeof order $0<\alpha \le 1$, where $A$ is a self-adjoint and bounded aboveoperator on a Hilbert space $H$. First, we extend the logarithmic convexitytechnique to the fractional framework by analyzing the properties of theMittag-Leffler functions. Then we prove conditional stability estimates ofH\"older type for initial conditions under a weaker norm of the final data.Finally, we give several applications to show the applicability of our abstractresults.

The backward problem for time fractional evolution equations