Reweighting average of increasing function












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Consider the entity $A=frac{int_a^b w(x)xf(x)dmu(x)}{int_a^b w(x)xdmu(x)}$, where $f$ is a smooth, monotonously increasing function with positive values: $f,f^prime>0$, $mu$ is the measure used in the integration, and $w$ is a smooth function with positive values: $w>0$. The derivative of $w$ is assumed to have the same sign everywhere: $w^prime=0$ everywhere, $w^prime>0$ everywhere or $w^prime<0$ everywhere.



Now consider the entity $B=frac{int_a^b frac{x}{w(x)}f(x)dmu(x)}{int_a^b frac{x}{w(x)}dmu(x)}$.



I have a strong intuition that the sign of $w^prime$ determines which is greater, $A$ or $B$. If $w^prime=0$ everywhere, then clearly $A=B$.



Now my intuition tells me the following:



$Alessgtr BLeftrightarrow w^primelessgtr 0$.



The intuition is clear: given that $f$ increases, then if $w$ increases, $A$ gives a higher weight to larger values of $f$ than $B$ does. If $w$ decreases, $A$ gives a lower weight to larger values of $f$ than $B$ does.



Only, I haved tried for a long time, I am unable to prove this intuition more formally. Is my intuition correct, and why? Or is my intuition wrong, and is there an easy counterexample?










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    $begingroup$


    Consider the entity $A=frac{int_a^b w(x)xf(x)dmu(x)}{int_a^b w(x)xdmu(x)}$, where $f$ is a smooth, monotonously increasing function with positive values: $f,f^prime>0$, $mu$ is the measure used in the integration, and $w$ is a smooth function with positive values: $w>0$. The derivative of $w$ is assumed to have the same sign everywhere: $w^prime=0$ everywhere, $w^prime>0$ everywhere or $w^prime<0$ everywhere.



    Now consider the entity $B=frac{int_a^b frac{x}{w(x)}f(x)dmu(x)}{int_a^b frac{x}{w(x)}dmu(x)}$.



    I have a strong intuition that the sign of $w^prime$ determines which is greater, $A$ or $B$. If $w^prime=0$ everywhere, then clearly $A=B$.



    Now my intuition tells me the following:



    $Alessgtr BLeftrightarrow w^primelessgtr 0$.



    The intuition is clear: given that $f$ increases, then if $w$ increases, $A$ gives a higher weight to larger values of $f$ than $B$ does. If $w$ decreases, $A$ gives a lower weight to larger values of $f$ than $B$ does.



    Only, I haved tried for a long time, I am unable to prove this intuition more formally. Is my intuition correct, and why? Or is my intuition wrong, and is there an easy counterexample?










    share|cite|improve this question











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      0








      0





      $begingroup$


      Consider the entity $A=frac{int_a^b w(x)xf(x)dmu(x)}{int_a^b w(x)xdmu(x)}$, where $f$ is a smooth, monotonously increasing function with positive values: $f,f^prime>0$, $mu$ is the measure used in the integration, and $w$ is a smooth function with positive values: $w>0$. The derivative of $w$ is assumed to have the same sign everywhere: $w^prime=0$ everywhere, $w^prime>0$ everywhere or $w^prime<0$ everywhere.



      Now consider the entity $B=frac{int_a^b frac{x}{w(x)}f(x)dmu(x)}{int_a^b frac{x}{w(x)}dmu(x)}$.



      I have a strong intuition that the sign of $w^prime$ determines which is greater, $A$ or $B$. If $w^prime=0$ everywhere, then clearly $A=B$.



      Now my intuition tells me the following:



      $Alessgtr BLeftrightarrow w^primelessgtr 0$.



      The intuition is clear: given that $f$ increases, then if $w$ increases, $A$ gives a higher weight to larger values of $f$ than $B$ does. If $w$ decreases, $A$ gives a lower weight to larger values of $f$ than $B$ does.



      Only, I haved tried for a long time, I am unable to prove this intuition more formally. Is my intuition correct, and why? Or is my intuition wrong, and is there an easy counterexample?










      share|cite|improve this question











      $endgroup$




      Consider the entity $A=frac{int_a^b w(x)xf(x)dmu(x)}{int_a^b w(x)xdmu(x)}$, where $f$ is a smooth, monotonously increasing function with positive values: $f,f^prime>0$, $mu$ is the measure used in the integration, and $w$ is a smooth function with positive values: $w>0$. The derivative of $w$ is assumed to have the same sign everywhere: $w^prime=0$ everywhere, $w^prime>0$ everywhere or $w^prime<0$ everywhere.



      Now consider the entity $B=frac{int_a^b frac{x}{w(x)}f(x)dmu(x)}{int_a^b frac{x}{w(x)}dmu(x)}$.



      I have a strong intuition that the sign of $w^prime$ determines which is greater, $A$ or $B$. If $w^prime=0$ everywhere, then clearly $A=B$.



      Now my intuition tells me the following:



      $Alessgtr BLeftrightarrow w^primelessgtr 0$.



      The intuition is clear: given that $f$ increases, then if $w$ increases, $A$ gives a higher weight to larger values of $f$ than $B$ does. If $w$ decreases, $A$ gives a lower weight to larger values of $f$ than $B$ does.



      Only, I haved tried for a long time, I am unable to prove this intuition more formally. Is my intuition correct, and why? Or is my intuition wrong, and is there an easy counterexample?







      real-analysis






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      share|cite|improve this question













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      edited Jan 8 at 13:20







      Scindapsus

















      asked Jan 8 at 13:14









      ScindapsusScindapsus

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