Positive integer solutions to $\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{x+y}=4$ I was wondering if anyone knew of any positive integer solutions to this Diophantine equation, or had a proof there are none. Integer solutions exist with negative values, such as (11,9,-5) and (4,11,-1), but checking positive integers up through 10,000 yielded nothing and I don't see a way to show there are none.
What are $x$, $y$ and $z$ if $\frac{x}{y + z} + \frac{y}{x + z} + \frac{z}{x + y} = 4$ and $x$, $y$ and $z$ are whole numbers? What are $x$, $y$ and $z$ if $$\dfrac{x}{y + z} + \dfrac{y}{x + z} + \dfrac{z}{x + y} = 4$$ and $x$, $y$ and $z$ are whole numbers?  MY ATTEMPT  Let $u = x + y + z$.  Then the equation can be rewritten as $$\dfrac{x}{u - x} + \dfrac{y}{u - y} + \dfrac{z}{u - z} = 4$$  Suppose I set $$1 = \dfrac{x}{u - x} = \dfrac{y}{u - y}$$ and $$2 = \dfrac{z}{u - z}.$$  Then I get $$x = y + z$$ $$y = x + z$$ $$z = 2(x + y),$$ so that $$z = 0 = x + y,$$ which is impossible.  Next, suppose I set $$\dfrac{4}{3} = \dfrac{x}{u - x} = \dfrac{y}{u - y} = \dfrac{z}{u - z}.$$  Then I get $$4(u - x) = 3x$$ $$4(u - y) = 3y$$ $$4(u - z) = 3z$$ so that $$12u - 4(x + y + z) = 3(x + y + z)$$ which implies that $$12u = 7(x + y + z) = 7u$$ from which it follows that $$u = 0.$$ This is, again, impossible.  Alas, here is where I get stuck.  Any hint(s) will be appreciated.
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Suppose $\phi$ is a real function on $\mathbb{R}$ such that $$\phi \Bigl(\int_0^1f(x)\,dx\Bigr)\leq \int_0^1 \phi(f)\,dx$$ for every real bounded measurable function $f$. Prove that $\phi$ is then convex.    This one is in Rudin's Real and Complex Analysis, page 74, and, to be quite honest, I am unsure what my first step to solve this problem should even be. To help think of a general methodology, I attempted this similar question, from page 71 of the same book:      Assume $\phi$ is a continuous real valued function on $(a,b)$ such that $$\phi\Bigl(\frac{x+y}{2}\Bigr)\leq\frac{\phi(x)+\phi(y)}{2}$$ for all $x, y\in (a,b).$ Prove that $\phi$ is then convex (the conclusion does not follow if continuity is omitted from the hypothesis).   I take it that the latter question is just a special case of the former (assuming $\phi$ is bounded on $(a,b)$), so I attempted to solve the latter problem first. 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