User talk:Saubhik Mukherjee

Very Very Brief list of some handy inequalities for a forgetful amateur problem solver
Inequalities

Theorem. The quadratic function is always positive; i.e.,. By substituting different expressions for, many of the inequalities below are obtained.Theorem. Bernoulli 's inequalities 1. If is an integer and  a real number then. 2. If or  then for  the following inequality holds:. 3. If then for  the following inequality holds: .Theorem. The mean inequalities For positive real numbers   it follows that, where

Each of these inequalities becomes an equality if and only if. The numbers, , , and are respectively called the quadratic mean, the arithmetic mean, the geometric mean, and the harmonic mean of .Theorem. The general mean inequality. Let be positive real numbers. For each we define the mean of order  of  by  for, and for. In particular,, , , , , and are , , , , , and respectively. Then whenever Theorem. Cauchy-Schwarz inequality. Let, , be real numbers. Then

Equality occurs if and only if there exists such that  for .Theorem. Holder 's inequality. Let, , be nonnegative real numbers, and let be positive real numbers such that. Then

Equality occurs if and only if there exists such that  for. The Cauchy--Schwarz inequality is a special case of Holder 's inequality for.Theorem. Minkowski 's inequality. Let  be nonnegative real numbers and  any real number not smaller than. Then

For equality occurs if and only if there exists  such that  for. For equality occurs in all cases.Theorem. Chebyshev 's inequality. Let and  be real numbers. Then

The two inequalities become equalities at the same time when or .Definition. A real function defined on an interval  is convex, if  for all  and all  such that. A function is said to be concave if the opposite inequality holds, i.e., if  is convex.Theorem. If is continuous on an interval, then  is convex on that interval if and only if  for all Theorem. If is differentiable, then it is convex if and only if the derivative  is nondecreasing. Similarly, differentiable function is concave if and only if  is nonincreasing.Theorem. Jensen 's inequality If is a convex function, then the inequality

holds for all, , and. For a concave function the opposite inequality holds.Theorem. Muirhead 's inequality Given and an -tuple  of positive real numbers, we define

the sum being taken over all permutations of. We say that an -tuple majorizes an -tuple  if  and for each. If a nonincreasing -tuple majorizes a nonincreasing -tuple, then the following inequality holds:

Equality occurs if and only if .Theorem. Schur 's inequality Using the notation introduced for Muirhead 's inequality,

where. Equality occurs if and only if or,  (and in analogous cases).