A new improved Jensen like inequality with double integral

It’s studied the improvement problem of Jensen like inequality with double integral.The integrand function in the new inequality we improved is a quadratic function excluding derivative.Firstly,the more concise and clear proof methods is provided through applying construction method of functions on the proof of this type of existing Jensen inequality in this article.Secondly,based on the above method of proof,a new type Jensen like inequality with double integral is derived by combining with free matrix thought and calculating technics of integral inequality.Finally,the reasonability,superiority and less conservative of the proposed new inequalities are analyzed theoretically.