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How to establish regions of convexity/concavity of a ratio of exponential polynomials?

Problem: Let $f\colon \mathopen[0,1\mathclose] \to \mathbb{R}$ be defined as $$ f(x) = \frac{e^{\rho x}-1}{e^{\rho x}-1+e^{\rho (1-\gamma x)}-e^{\rho (1-\gamma) x}} $$ where $\rho$ and $\gamma$ are st …
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