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To answer this question first i derived the Hessian matrix of $f(x,y)=-\ln(1+\frac{x}{by+a})$ which is as follows:

$$\triangledown^2 f(x,y)=\frac{1}{(by+x+a)^2} \begin{bmatrix} 1 & b \\b & -\frac{b^2x(2by+x+2a)}{(by+a)^2} \end{bmatrix}$$ This matrix is not positive semi definite. Accordingly, i cannot say it is concave function.

On the other hand, $g(x,y)=\frac{x}{by+a}$ is linear-fractional function and $\ln(x)$ is concave function and the composition of a concave function with linear-fractional function is a concave function. Therefore, $f(x,y)=\ln(1+\frac{x}{by+a})$ is concave.

So which one is true, and if it is not concave, is it a non convex problem?

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    "the composition of a concave function with linear-fractional function is a concave function" who said this? – 2017-02-07
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    @GGG thank you for your comment. In fact you are right, the composition of a concave function with affine function and not Linear-fractional function is a concave function. Do you know if it is quasi-concave function, because the shape of this equation has some concavity. In other word, can i use the Lagrangian method to solve such optimization problem. – 2017-02-08

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