Condition: $$E[\text{max}(X,Y)] \leq E[\text{max}(K,Y)]$$
Here, $X,Y$ are random variables. $K$ is a constant. The distribution for $Y$ is known.
Question one:
Is it possible to find the distribution parameters for $X$ satisfying the above condition?
Question two:
Is it possible to find the distribution parameters for $X$ satisfying the above condition when $X,Y$ are non negative random variables and $K \geq 0$ ?
Clarifications:
Please note,
1) $X,Y$ can follow any probability distribution in the above two questions.
2) So the problem comes down to assuming a particular distribution for $X$ and finding the parameters of the distribution satisfying the above condition.
3) We can make other simplifying assumptions if required, such as $X$, $Y$ are independent.
Special Case Related Question:
Please note, the special case of finding the maximum of two independent log normally distributed random variables, which is required for a part of the answer for this present question, is considered here: Expected Value of Maximum of Two Lognormal Random Variables