Concave Inequalities

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I would like to ask if Popoviciu's inequality holds for concave functions but with a change of the inequality sign to its opposite.



And generally if for known inequalities about convex functions can we just consider concave functions instead of convex functions and change the inequality sign to its opposite ( when we have ">" for convex functions then use "<" instead for concave functions) then the new inequality still holds for the concave functions?



That is, is there a duality property for inequalities among convex and concave functions?



Thanks in advance.




convex-analysis




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  • Just apply the inequality to the convex function $-f$.
    – LinAlg
    Jul 21 at 1:40














up vote
0
down vote

favorite












I would like to ask if Popoviciu's inequality holds for concave functions but with a change of the inequality sign to its opposite.



And generally if for known inequalities about convex functions can we just consider concave functions instead of convex functions and change the inequality sign to its opposite ( when we have ">" for convex functions then use "<" instead for concave functions) then the new inequality still holds for the concave functions?



That is, is there a duality property for inequalities among convex and concave functions?



Thanks in advance.




convex-analysis




share|cite|improve this question



















  • Just apply the inequality to the convex function $-f$.
    – LinAlg
    Jul 21 at 1:40












up vote
0
down vote

favorite









up vote
0
down vote

favorite











I would like to ask if Popoviciu's inequality holds for concave functions but with a change of the inequality sign to its opposite.



And generally if for known inequalities about convex functions can we just consider concave functions instead of convex functions and change the inequality sign to its opposite ( when we have ">" for convex functions then use "<" instead for concave functions) then the new inequality still holds for the concave functions?



That is, is there a duality property for inequalities among convex and concave functions?



Thanks in advance.




convex-analysis




share|cite|improve this question











I would like to ask if Popoviciu's inequality holds for concave functions but with a change of the inequality sign to its opposite.



And generally if for known inequalities about convex functions can we just consider concave functions instead of convex functions and change the inequality sign to its opposite ( when we have ">" for convex functions then use "<" instead for concave functions) then the new inequality still holds for the concave functions?



That is, is there a duality property for inequalities among convex and concave functions?



Thanks in advance.





convex-analysis





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 20 at 14:45









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  • Just apply the inequality to the convex function $-f$.
    – LinAlg
    Jul 21 at 1:40
















  • Just apply the inequality to the convex function $-f$.
    – LinAlg
    Jul 21 at 1:40















Just apply the inequality to the convex function $-f$.
– LinAlg
Jul 21 at 1:40




Just apply the inequality to the convex function $-f$.
– LinAlg
Jul 21 at 1:40















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