Inequalities for a class of bounded functions (Q2730586)

The author introduces the functional space \(M_\varphi [0,1 ]:= \{ f\in M[0,1 ]\mid f\geq \varphi\); \(\varphi(0)=\varphi_0\), \(\varphi(1) =\varphi_1\},\) where \(M[0,1 ]\) denotes the space of bounded real functions on \([0,1 ]\), and \(\varphi\) is a fixed, convex function on \([0,1 ]\). Two inequalities are derived for the centroid of a polygon \(P_n=A_1\dots A_n\); \(A_i\in f\in M_\varphi [0,1 ]\). Denote \(t^{[p ]}_n := \sum_{i=1}^np_it_i\) the weighted \(p\)-mean of the \(n\)-tuple \((t_1,\cdots,t_n)\) with respect to the weights \(p_i\geq 0\); \(\sum_{i=1}^np_i=1\), when \(t_i\) belong to \([0,1 ]\). Then for \(t_i, \tau\in [0,1 ]\), \(f\in M_\varphi[0,1 ]\) it is NEWLINE\[NEWLINE \sum_{i=1}^np_if(t_i)\geq \varphi (\tau), \tag{1} NEWLINE\]NEWLINE even if \(t^{[p ]}_n=\tau\). When \(\varphi\) is decreasing on \([t^{[p ]}_n,1 ]\), then (1) holds for all \(t_i\in [0,1 ]\), \(i=\overline{1,n}\), such that \(t^{[p ]}_n\leq \tau\); when \(\varphi\) is increasing on \([0,t^{[p ]}_n ]\), then (1) is true for \(t^{[p ]}_n\geq \tau\). Let \(f\in M_s[0,1 ]\), \(t,t_i,\tau \in [0,1 ]\) and assume that \(\varphi_0\leq \tau M_f\leq \varphi_1\); \(M_f = \sup_{[0,1 ]}f(t)\). Then for \(t_n^{[p ]}\in [\frac{\tau M_f-\varphi_0}{\varphi_1-\varphi_0},1]\) we have that NEWLINE\[NEWLINE \sum_{i=1}^np_if(t_i)\geq \tau M_f \geq \tau f(t). NEWLINE\]NEWLINE When \(\varphi_1\leq \tau M_f\leq \varphi_0\), the same inequality holds if \(t_n^{[p ]}\in [0,\frac{\tau M_f-\varphi_0}{\varphi_1-\varphi_0}]\). These inequalities are sharpened with respect to the ``guard''-function \(\varphi\) with assumed, known endpoints \(\varphi(0)=\varphi_0, \varphi(1) =\varphi_1\). Finally two illustrative examples are discussed.