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$$f(x,y)=-x\log(1+y)$$ The Hessian matrix of $f(x,y)$ is $$\left[ \begin{matrix} 0 & -\frac{1}{1+y}\\ -\frac{1}{1+y} & \frac{x}{(1+y)^2} \end{matrix}\right] $$

Then the eigen value is $-\frac{1}{(1+y)^2}$.

Since it is not positive, $f(x,y)$ is not convex. Right??

However, papers said it is convex.

Please let me know why it is convex.

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    It is not convex. Are you sure you didn't misunderstand what the papers say? What are the papers you talk about? – 2017-01-26
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    Hmm... ah not convex?? I am correct?? thank you for giving me confidence. I will check again the paper. So hard to understand it sorry for bad question.... – 2017-01-26

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