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