## Class 6 Notes

24 sept 2018 log x. A more refined answer: it looks like a certain integral called ... Proof (Ben): The derivative of logx is 1 x and the derivative of.

LS

## 1 Theory of convex functions

1 mar 2016 Strongly convex if ∃α > 0 such that f(x) − α

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## Chapter 8 Logarithms and Exponentials: logx and e

Exercise 4 Prove the Laws of Exponents. Hint: make use of the fact that log x is a 1-1 function. Derivatives of the Exponential Function. We already

chapter

## New sharp bounds for the logarithmic function

5 mar 2019 In this paper we present new sharp bounds for log(1 + x). We prove that our upper bound is sharper than all the upper bounds presented ...

## Proof of the Sheldon Conjecture

13 feb 2019 x log x for all x ≥ 17. (1). January 2014]. PROOF OF THE SHELDON CONJECTURE ... Letting z = log x this derivative is z + log z + 1/z.

sheldon

## Some Inequalities involving ( r!)1/ r

Lemma 1. If x> 1 then. 0<log (r(x))-{(x-$) log (x)-x+i log (2«). Proof. Proof. We prove that for JC^6 the second derivative of h(x) is negative.

div class title some inequalities involving span class italic r span span class sup span class italic r span span div

## Chapter 3 Elementary Prime Number Theory

log p ps . Thus without justification of proving that the derivative of a series Denote the sum over the logarithm of primes by θ (x) = ∑ p≤x log p.

Notes

## CONTINUITY AND DIFFERENTIABILITY

log. = x b. 6. logb b = 1 and logb 1 = 0. (iv) The derivative of ex Example 9 If ex + ey = ex+y prove that ... Example 13 If xy = ex–y

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## 1 Fisher Information

6 abr 2016 log f(X

Fisher

## 3. Convex functions

zk) (from Cauchy-Schwarz inequality) geometric mean: f(x)=(∏ n k=1 xk). 1/n on R n. ++ is concave. (similar proof as for log-sum-exp). Convex functions.

functions