There are $n$ ($n \ge 3$) independent random variables $\{ {c_i}\} _{i = 1}^n$ identically drawn on the interval $[\underline c,\bar c]$ ($\underline c>0$), with cdf $F(\cdot)$ and pdf $f(\cdot)$, where $\frac{{F(\cdot)}}{{f(\cdot)}}$ is increasing. Denote ${S_i} = 1 - \frac{{\psi ({c_{(i)}})}}{r}$ and ${V_i} = 1 - \frac{{{c_{(i)}}}}{r}$ , where $\psi (c) = c + \frac{{F(c)}}{{f(c)}}$ , and ${c_{(i)}}$ is the $i$-th smallest order statistics. Denote $X = V_3^2$ and $Y = \frac{{(1 + \theta)S_1^2}}{2} + \frac{{{{[{{({S_2} - \theta {S_1})}^ + }]}^2}}}{{2(1 - \theta )}}$, where $0<\theta<1$ and ${(x)^ + } = \left\{ {\begin{array}{*{20}{c}} x&{if{\kern 1pt} x > 0}\\ 0&{if{\kern 1pt} x \le 0} \end{array}} \right.$. Let ${\mu _Y}(n)$ and ${\mu _X}(n)$ be the expected values of $X$ and $Y$, respectively. Try to show that $T(n) = \frac{{{\mu _Y}(n)}}{{{\mu _X}(n)}}$ is decreasing in $n$, which has been verified by numerical simulation but not been proved analytically.
Notes: The following results may be useful for the proof.
(1) It has been proved that ${\mu _Y}(n) > {\mu _X}(n)$ holds always.
(2) Denote $\bar F(c) = 1 - F(c)$. The pdf of ${c_{(i)}}$ can be written as ${f_{(i)}}(x) = \frac{{n!}}{{(i - 1)!(n - i)!}}{F^{i - 1}}(x){\bar F^{n - i}}(x)f(x)$, and the joint density function of $({c_{(1)}},{c_{(2)}})$ is ${f_{(1)(2)}}({x_1},{x_2}) = n(n - 1){\bar F^{n - 2}}({x_2})f({x_1})f({x_2})$.
(3) One can show that $2E[{V_3}] = E[{S_1}] + E[{S_2}]$.
(4) ${\mu _X}(n) = {(1 - \frac{{\underline c }}{r})^2} \\
{\kern 35pt} + \int_{\underline c }^{\bar c} {{V_3}{V_3}^\prime {{\bar F}^{n - 2}}(x)[(n - 1)(n - 2){{\bar F}^2}(x) - 2n(n - 2)\bar F(x) + n(n - 1)]dx}, $
${\mu _Y}(n) = {(1 - \frac{{\underline c }}{r})^2} + \int_{\underline c }^{\bar c} {{S_1}{S_1}^\prime {{\bar F}^{n - 1}}(x)[2\bar F(x) + nF(x)]dx} \\
{\kern 35pt} + \frac{{n\theta }}{{1 - \theta }}\int_{\underline c }^{\bar c} {\int_{{L_1}({x_2})}^{{x_2}} {{S_1}^\prime {S_2}^\prime {{\bar F}^{n - 1}}({x_2})F({x_1})d{x_1}d{x_2}} }.$
(5) ${\mu '_X}(n) = {\mu _X}(n) - {\mu _X}(n - 1) \\ {\kern 28pt} = - (n - 1)(n - 2)\int_{\underline c }^{\bar c} {{V_3}{{V'}_3}} {\bar F^{n - 3}}(x){F^3}(x)dx > 0,$ $\begin{array}{l} {{\mu '}_Y}(n) = {\mu _Y}(n) - {\mu _Y}(n - 1) \\ {\kern 28pt} = \int_{\underline c }^{\bar c} {{S_2}{S_2}^\prime {{\bar F}^{n - 2}}(x)F(x)[(n - 2)\bar F(x) - (n - 1)]dx} \\ {\kern 38pt} + \frac{\theta }{{1 - \theta }}\int_{\underline c }^{\bar c} {\int_{{L_1}({x_2})}^{{x_2}} {{S_1}^\prime {S_2}^\prime {{\bar F}^{n - 2}}({x_2})F({x_1})[n\bar F({x_2}) - (n - 1)]d{x_1}d{x_2}}. } \end{array}$
(6) $T'(n) = T(n) - T(n - 1) = \frac{{{{\mu '}_Y}(n) - T(n){{\mu '}_X}(n)}}{{{\mu _X}(n - 1)}} < \frac{{{{\mu '}_Y}(n) - {{\mu '}_X}(n)}}{{{\mu _X}(n - 1)}}$. So one can conclude the proof if ${\mu '_Y}(n) < {\mu '_X}(n)$.
