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I am trying to figure out the following: enter image description here

I know I have not shown my attempt but trust me I have used everything I can think of to get it right. For example, first expanding the numerator in the fraction on RHS in the top equation. Taking it on the other side and factoring out $$\frac{1}{{c_1}^a*}$$ from both sides. However get stuck from there onwards.

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  • $\begingroup$ the equation is correct, but the inequality for the convex sum $\theta + (1-\theta) \rho$ depends on the sizes and signs of both $\theta, \rho.$ The requirement is $0 < \theta + (1-\theta) \rho < 1$ $\endgroup$
    – Will Jagy
    Apr 27, 2020 at 2:51

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Hint: Clear denominators by multiplying both sides by $c_1^{a*}(1-\theta c_1^{a*})$. Then solve the resulting linear equation for $c_1^{a*}$.

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  • $\begingroup$ Brilliant tip. I figured it out. $\endgroup$
    – user508281
    Apr 27, 2020 at 3:24

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