We prove that the Arimoto-Blahut like algorithm provided by Yasui et al [6] to solve for the Secrecy Capacity of a less-noisy Discrete Memoryless Channel (DMC) pair can be extended to a more-capable DMC pair subject to the availability of a suitable initial guess. In particular, we show that if a cut parallel to the input hyperplane removes a convex piece from the graph of the multi-input function f(q) = I(X;Y) - I(X: Z) q(x), where q is the input probablity distribution, then for any initial guess chosen within that convex piece, Yasui's algorithm will converge to the optimal value in that convex piece. We then introduce a new characterization called quasiconcavity of a DMC pair and show that it lies between the less-noisy and more-capable characterizations. We also show that in the binary case, quasiconcavity is equivalent to the more-capable characterization. We then show that we can choose from a wider range of initial guesses by looking at regions around the optimal point where the function is quasiconcave. Finally we establish that the algorithm of Yasui et al can be used for more-capable channel pairs with binary alphabets. © 2008 IEEE.