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2. Lebedev–Milin inequality - Wikipedia

   en.wikipedia.org/wiki/Lebedev–Milin_inequality

   In mathematics, the Lebedev–Milin inequality is any of several inequalities for the coefficients of the exponential of a power series, found by Lebedev and Milin and Isaak Moiseevich Milin . It was used in the proof of the Bieberbach conjecture , as it shows that the Milin conjecture implies the Robertson conjecture .

3. Bernoulli's inequality - Wikipedia

   en.wikipedia.org/wiki/Bernoulli's_inequality

   Bernoulli's inequality can be proved for case 2, in which is a non-negative integer and , using mathematical induction in the following form: we prove the inequality for {,}, from validity for some r we deduce validity for +.

4. List of inequalities - Wikipedia

   en.wikipedia.org/wiki/List_of_inequalities

   Bennett's inequality, an upper bound on the probability that the sum of independent random variables deviates from its expected value by more than any specified amount Bhatia–Davis inequality , an upper bound on the variance of any bounded probability distribution

5. Young's inequality for products - Wikipedia

   en.wikipedia.org/wiki/Young's_inequality_for...

   Proof [2]. Since + =, =. A graph = on the -plane is thus also a graph =. From sketching a visual representation of the integrals of the area between this curve and the axes, and the area in the rectangle bounded by the lines =, =, =, =, and the fact that is always increasing for increasing and vice versa, we can see that upper bounds the area of the rectangle below the curve (with equality ...

6. Inequality (mathematics) - Wikipedia

   en.wikipedia.org/wiki/Inequality_(mathematics)

   For instance, to solve the inequality 4x < 2x + 1 ≤ 3x + 2, it is not possible to isolate x in any one part of the inequality through addition or subtraction. Instead, the inequalities must be solved independently, yielding x < ⁠ 1 / 2 ⁠ and x ≥ −1 respectively, which can be combined into the final solution −1 ≤ x < ⁠ 1 / 2 ⁠.

7. Abel's theorem - Wikipedia

   en.wikipedia.org/wiki/Abel's_theorem

   As an immediate consequence of this theorem, if is any nonzero complex number for which the series = converges, then it follows that = = in which the limit is taken from below.