@Gerry Myerson - thank you. May somebody help with a paper in Bulletin des Sciences Mathematiques? A first paper-not an easy reading...

@Vladimir Zolotov - pochtaname(c)gmail.com

@Lev Borisov - I solved it by Lagrange. The aim of the question is to find geometrical solution using convexity of related surface.

Do you know a reference to Ostrowski's paper? Is not a first proposed inequality a consequence?

Yes, it is exactly what I wanted, user35593. Thank you very much!

Lev Borisov - I proved the underlying inequality of 3rd degree as a lemma for some result in joint paper. But the proof is not nice or short. So I am interesting to find another one using geometry and convexity.

Somebody of the usual set "put on hold" may prove it? Will be thankful...

We may take an integral using a technique based on Slater's Theorem (cf. O.I.Marichev's book). In fact it will use in an inderect way contour integrals and residues.

Is not the question we consider then equivalent that $grad g =O$? More: $rot(rot F)=O => rot F =grad g $ for some g. Question: when this $rot F=O$ ? Answer: only if g=const.

Is not the question we consider then equivalent that $grad g =O$?

$rot (grad g)=O (0-vector)$, is not it?

The main problem and the pity of this site is that many participants are too fast and not objective with their "ON HOLD" decision and demonstrating "WANT TO BE CLEVER THEN OTHERS" position. Say for not simply connected domains this is an interesting question, I do not know is it solved or not.