What are the rules for using maximum in inequalities?

[Niles]

Hi guys

Today my lecturer wrote on the blackboard

<br /> \max \left\{ {\alpha f(x) + (1 - \alpha )f(y)\,\,,\,\,\alpha g(x) + (1 - \alpha )g(y)} \right\}\,\,\,\, \le \,\,\,\alpha \max \left\{ {f(x)\,\,,\,\,g(x)} \right\} + (1 - \alpha )\max \left\{ {f(y)\,\,,\,\,g(y)} \right\},<br />

where x, y are variables in all R, and alpha is a constant in [0;1]. I must admit, I cannot quite see why this inequality holds. Are there some rules about the maximum that is being used here?

 

[Lord Crc]

Ok, starting with

\max \left\{ \alpha f(x) + (1 - \alpha )f(y),\alpha g(x) + (1 - \alpha )g(y) \right\}

Obviously, f(x) \le \max \left\{f(x), g(x)\right\} = x_{max}, and similar for g(x), so we can safely use this to get something which is greater or equal:

\max \left\{ \alpha f(x) + (1 - \alpha )f(y),\alpha g(x) + (1 - \alpha )g(y) \right\} \le \max \left\{ \alpha x_{max} + (1 - \alpha )f(y),\alpha x_{max} + (1 - \alpha )g(y) \right\}Again, f(y) \le \max \left\{f(y), g(y)\right\} = y_{max}, and similar for g(y), and using that we get

\max \left\{ \alpha x_{max} + (1 - \alpha )f(y),\alpha x_{max} + (1 - \alpha )g(y) \right\} \le \max \left\{ \alpha x_{max} + (1 - \alpha )y_{max},\alpha x_{max} + (1 - \alpha )y_{max} \right\}Now, for a \ge 0 we have that \max \left \{ ab, ac \right \} = a \max \left \{b, c \right \} and \max \left \{ a + b, a + c \right \} = a + \max \left \{b, c \right \}. Assuming \alpha \in [0,1], we can use this to get

\max \left\{ \alpha x_{max} + (1 - \alpha )y_{max},\alpha x_{max} + (1 - \alpha )y_{max} \right\} = \alpha x_{max} + (1 - \alpha )\max \left\{y_{max}, y_{max} \right\} = \alpha x_{max} + (1 - \alpha )y_{max}.

Inserting the expressions for x_{max} and y_{max} gets you to the right hand side you had, which thus is equal or greater than the left hand side.

 

[Niles]

Thank you. It is very kind of you to help.