While finding maxima or minima of some special functions, we use the idea of the arithmetic mean being greater than the geometric mean. This makes simplification easier and we can find maxima or minima of a certain function easily (compared to differentiating it).
But we know that the geometric mean is greater than the harmonic mean, does it mean that, if I use the inequality(GM>HM OR HM<AM) in certain functions, will i get a higher maxima or lower minima? If not, why?
Clarification -We can find maxima or minima of certain special function using the result AM>GM. But we know that AM>GM>HM so will i get better bounds if i use AM>HM rather than AM>GM. Please give a generalized answer not for a specific examples.
The question is vague as it refers to "certain functions".
If what is meant by the question is that the inequality $AM>HM$ should give a better estimate than $AM>GM$ (since $GM>HM$), then the answer is normally not.
In general $$AM\ge GM,\qquad AM\ge HM = \frac{HM}{GM}GM$$ Since $HM/GM\le1$, one can see that the second inequality is less sharp than the first, in the sense that the first one implies the second. So anything that comes out of the second inequality can also be shown using the first.
But there are many cases, where the two are equivalent. For example, take $F=a+b$, then $AM(a,b)=F/2$, $GM(a,b)=\sqrt{ab}$, $HM(a,b)=2ab/F$.
Hence $GM\le AM$ gives $F\ge2\sqrt{ab}$.
$HM\le AM$ gives $F/2\ge2ab/F$, which is the same inequality $F\ge2\sqrt{ab}$.