Abstract:
In 1957 Ky Fan gave a necessary and sucient condition, known as Fan's Consistency Condition, for a finite system of convex in-equalities to have a solution. This result has been somewhat overshadowed by the famous Fan's Inequality which is equivalent to Brouwer's Fixed Point Theorem. Another result which bears Fan's name, but which is not due to him, is Fan's Lopsided Inequality which Aubin and Ekeland prove in [1] using Fan's Inequality.We first prove a fairly general, but elementary result, Theorem 2.1.1, from which we derive both Fan's Theorem for nite systems of convex inequalities and Fan's Lopsided Inequality whose proof, therefore, does not require Brouwer's Fixed Theorem. We show that Theorem 2.1.1 is equivalent to Fan's Theorem for nite systems of convex inequalities; consequently, the Lopsided Inequality is a consequence of Fan's Theorem for finite systems of convex inequalities.A number of well known and important results are proved along theway. The paths leading from Fan's 1957 theorem to those results are,we hope, simple enough to demonstrate that it deserves to be as wellknown as its younger and powerful cousin, Fan's Inequality.

Abstract:
There are quite a few generalizations or applications of the 1984 minimax inequality of Ky Fan compared with his original 1972 minimax inequality. In a certain sense, the relationship between the 1984 inequality and several hundreds of known generalizations of the original 1972 inequality has not been recognized for a long period. Hence, it would be necessary to seek such relationship. In this paper, we give several generalizations of the 1984 inequality and some known applications in order to clarify the close relationship among them. Some new types of minimax inequalities are added. 1. Introduction The KKM theory is originated from the Knaster-Kuratowski-Mazurkiewicz (KKM for short) theorem of 1929 [1]. Since then, it has been found a large number of results which are equivalent to the KKM theorem; see [2, 3]. Typical examples of the most remarkable and useful equivalent formulations are Ky Fan's KKM lemma of 1961 [4] and his minimax inequality of 1972 [5]. The inequality and its various generalizations are very useful tools in various fields of mathematical sciences. Since 1961, Ky Fan showed that the KKM theorem provides the foundation for many of the modern essential results in diverse areas of mathematical sciences. Actually, a milestone in the history of the KKM theory was erected by Fan in 1961 [4]. His 1961 KKM Lemma (or the Fan-KKM theorem) extended the KKM theorem to arbitrary topological vector spaces and had been applied to various problems in his subsequent papers [5–10]. Recall that, at the beginning, the basic theorems in the KKM theory and their applications were established for convex subsets of topological vector spaces mainly by Fan in 1961–1984 [4–10]. A number of intersection theorems and their applications to various equilibrium problems followed. In our previous review [11], we recalled Fan's contributions to the KKM theory based on his celebrated 1961 KKM lemma, and introduced relatively recent applications of the lemma due to other authors in the twenty-first century. Then, the KKM theory was extended to convex spaces by Lassonde in 1983 [12] and to -spaces (or H-spaces) by Horvath in 1983–1993 [13–16] and others. Since 1993, the theory has been extended to generalized convex (G-convex) spaces in a sequence of papers of the present author and others; see [2]. Since 2006, the main theme of the theory has become abstract convex spaces in the sense of Park [17–30]. The basic theorems in the theory have numerous applications to various equilibrium problems in nonlinear analysis and other fields. In our previous review [30], we

Abstract:
The aim of this paper is to analyze the weighted KyFan inequality proposed in [11]. A number of numerical simulations involving the exponential weighted function is given. We show that in several cases and types of examples one can imply an improvement of the standard KyFan inequality.

Abstract:
We prove an equivalent relation between Ky Fan-type inequalities and certain bounds for the differences of means. We also generalize a result of Alzer et al. (2001).

Abstract:
Under new assumptions, we provide suffcient conditions for the (upper and lower) semicontinuity and continuity of the solution mappings to a class of generalized parametric set-valued Ky Fan inequality problems in linear metric space. These results extend and improve some known results in the literature (e.g., Gong, 2008; Gong and Yoa, 2008; Chen and Gong, 2010; Li and Fang, 2010). Some examples are given to illustrate our results.

Abstract:
There are several inequalities in the literature carrying the name of Ky Fan. We survey these well-known Ky Fan inequalities and some other significant inequalities generalized by Ky Fan and review some of their recent developments.

Abstract:
We give some further consideration about logarithmic convexity for differences of power sums inequality as well as related mean value theorems. Also we define quasiarithmetic sum and give some related results.