सामान्यीकृत माध्य

यदि p एक अशून्य वास्तविक संख्या है, तथा  धनात्मक वास्तविक संख्याएँ हैं, तो इन संख्याओं का सामान्यीकृत माध्य (generalized mean) या p घात वाला घात माध्य (power mean) ' निम्नलिखित है-

 विशिष्ट स्थितियाँ 

{| class="toccolours collapsible collapsed" width="90%" style="text-align:left"
!Proof of  (geometric mean)
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We can rewrite the definition of Mp using the exponential function

In the limit p'' → 0, we can apply L'Hôpital's rule to the argument of the exponential function. Differentiating the numerator and denominator with respect to p, we have

By the continuity of the exponential function, we can substitute back into the above relation to obtain

as desired. 
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{| class="toccolours collapsible collapsed" width="90%" style="text-align:left"
!Proof of  and 
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Assume (possibly after relabeling and combining terms together) that . Then

The formula for  follows from 
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