In this paper, the main purpose is to estimate the upper bound of Sugeno fuzzy integral for log-convex functions using the classical Hadamard integral inequality.
In this paper, we give an upper bound for the Sugeno fuzzy integral of log-convex functions using the classical Hadamard integral inequality.
They introduced the idea by giving the general setting of the above inequality by using the classical majorization theorem for the function (f(x)= x log x), which is convex and continuous on (mathbf{R}_).
In [1], Cho et al. developed an iterative algorithm to approximate the solution of a system of nonlinear variational inequalities by using the classical resolvent operator technique.
First, it can be used to prove weak endpoint inequalities starting from strong-type inequalities, something which is impossible using the classical result.
If higher products are relevant, one must modify the classical inequality.
The main idea is to use the classical approach based on Riccati type inequality.
end{aligned} Using the Hölder inequality and the classical Trudinger Moser inequality, we have begin{aligned} int_{Omega }hu_{epsilon }^{2}e^{(4pi-epsilon )u_{epsilon }^{2}},dx leq e^{delta c_{epsilon }^{2}} int_{Omega }h u_{epsilon }^{2}e^{(4pi-epsilon -delta)u_{epsilon }^{2}},dx leq Ce^{delta c_{epsilon }^{2}}, end{aligned} where (0<delta<4pi), C depends only on h and δ.
In particular, our results extend some important inequalities in a classical situation; when (alpha=1 ), some relationships between these inequalities and the classical inequalities have been established.
Under the same conditions, there are the classical inequalities [2] (1.2). (1.3).
