So we're on a probability space equipped with a filtration $\mathcal F_t$. There are constants $\alpha,\ \beta,\ K\ >0$ and a stochastic process $X_t,\ t\leq T$ which consists of a $\mathcal F_t$ measurable part and a stochastic integral that has independent increments w.r.t. $\mathcal F_t$. I want to show the following: \begin{align} &\ \frac{1}{\alpha}\Big( \ln E\big[e^{\alpha X_T} \mid \mathcal F_t\big] - \ln E\big[e^{\alpha \min(K,X_T)}\mid \mathcal F_t\big]\Big)\\ \ge &\ E\big[X_T\mid \mathcal F_t]-E[\min(K,X_T)\mid \mathcal F_t\big]\\ \ge &\ - \frac{1}{\beta}\Big(\ln E\big[e^{-\beta X_T} \mid \mathcal F_t\big] - \ln E\big[e^{-\beta \min(K,X_T)}\mid \mathcal F_t\big]\Big) \end{align}
If the second term wasn't there, I could easily show this by Jensen's inequality. The Problem is the term which gets subtracted, which also becomes smaller if I use Jensen. Is there any theoretical result I can use here?
I also thought about a case analysis: If $K$ is the minimum, it doesnt really matter since it is not random. If $X_T$ is the minimum everything equates to zero. But I'm not really sure I am allowed to do it this way.
For reference these equations relate to indifference prices for options on the process $X_t$.
I appreciate any help on this.