I was wondering about the question in the title, as I knew some examples in the literature for proposed problems in journals or articles, about inequalities concerning the complex module or the inequalities for real numbers.

My motivation is my desire to know simple techniques to generate inequalities in two special cases. These inequalities apply to certain values of our (real or complex) variables.

The first example I invoke is (1) (in Spanish), for which I have tried to get a variant with the following identity as a starting point $$ log (1-z) sum_ {k = 0} ^ nz ^ k = log (1-z) cdot frac {z ^ {n + 1} -1} {z-1}. $$

The second nice example comes from (2), and I tried to create inequalities for real numbers by combining the authors' examples and Wikipedia, which is known as Fischer's inequality.

**Question.**

A) What simple techniques can be proposed to create examples of inequalities with the complex module? Feel free to illustrate it with an example of an original inequality.

B) What simple techniques can be proposed to produce examples of inequalities for real numbers, using a determinative inequality as a starting point? Feel free to illustrate it with an example of an original inequality.

**Many thanks.**

Therefore, I know the techniques of the following references and ask for other different techniques that give us similar inequalities. My experiments were artificial and I ask if I speak colloquially: about a professor's recipe for an amateur to create difficult inequalities with the complex module (question A) or to create difficult inequalities for real numbers (question B). Starting from an inequality of determinants.

## references:

(1) Oscar Ciaurri, *PROBLEM 24*, Section Problemas y Soluciones of La Gaceta de La RSME, Vol. 8.3 (2005), page 756.

(2) Daniel Sitaru and Leonard Giugiuc, *Application of Hadamard's Theorems*

inequalities, Section of Crux Mathematicorum, Vol. 44 (1), January 2018, Canadian Mathematical Society.