Let $\Phi : \mathbb{R}_{++}\to \mathbb{R}$ be a convex function (not necessarily differentiable). Fix an $\alpha \in (0,1)$ and define

$$g(t) = \alpha t .\Phi\left(\frac{1}{t} + 1\right) + 2(1-\alpha)t .\Phi\left(\frac{1}{t}+\frac{1}{2}\right),\quad \text{for }\;t\in[1,2].$$ Does there exist any such functions $\Phi$ such that $g(\cdot)$ is a constant as a function of $t$? I tried to tackle the problem using Jensen's inequality but nothing seems to work. For example, I have obtained $$\alpha t \Phi\left(\frac{1}{t} + 1\right) + 2(1-\alpha)t \Phi\left(\frac{1}{t}+\frac{1}{2}\right)\geq t(2-\alpha)\Phi((1+t)\alpha + (2+t)(1-\alpha)).$$ But this seems like it is not going to get me anywhere. A couple of facts that could be useful is that $t.\Phi\left(\frac{1}{t}+1\right)$ and $2t .\Phi\left(\frac{1}{t}+\frac{1}{2}\right)$, both are convex functions in $t$. Any help will be appreciated. Thanks.