# How to find the value in which an expression becomes greater than another

I have the following expressions:

$$\Delta \sigma_1=\frac{2 \gamma ^2 \left(\text{\sigma Y} \log \left(\frac{\gamma }{\alpha }\right)+\text{pe}\right)}{\left(\gamma ^2-1\right) \rho ^2}$$

$$\Delta \sigma_2=\frac{\frac{P}{\pi b^2}-\frac{\gamma ^2 \left(\rho ^2+1\right) \text{\sigma Y} \log \left(\frac{\gamma }{\alpha }\right)+\text{pe} \left(\gamma ^2+\rho ^2\right)}{\rho ^2}}{\gamma ^2-1}$$

$$\Delta \sigma_3=\frac{-\frac{\gamma ^2 \left(\rho ^2-1\right) \text{\sigma Y} \log \left(\frac{\gamma }{\alpha }\right)}{\rho ^2}+\frac{P}{\pi b^2}+\text{pe} \left(\frac{\gamma ^2}{\rho ^2}-1\right)}{\gamma ^2-1}$$

I need to find the greater between $$\Delta \sigma_1$$ and $$\Delta \sigma_2$$. Then I have to compare the max expression just found with $$\Delta \sigma_3$$, in order to find for which value of $$\gamma$$ the following inequality is satisfied:

max$$(\Delta \sigma_1$$, $$\Delta \sigma_2)\geq \Delta \sigma_3$$.

Note that: (P, pe, b, $$\sigma_Y$$, $$\alpha=a/b$$ are constant parameters; $$\rho=r/b$$ can be assumed as $$\rho=\gamma$$ and $$\gamma=c/b$$ is defined in $$]\alpha;1]$$

Thanks to everyone who will help me.

\[CapitalDelta]\[Sigma]el1 =(2 \[Gamma]^2 (pe + \[Sigma]Y Log[\[Gamma]/\[Alpha]]))/((-1 + \ \[Gamma]^2) \[Rho]^2)

\[CapitalDelta]\[Sigma]el2 =(P/(b^2 \[Pi]) - ( pe (\[Gamma]^2 + \[Rho]^2) + \[Gamma]^2 (1 + \[Rho]^2) \[Sigma]Y \ Log[\[Gamma]/\[Alpha]])/\[Rho]^2)/(-1 + \[Gamma]^2) 

\[CapitalDelta]\[Sigma]el3 =(P/(b^2 \[Pi]) + pe (-1 + \[Gamma]^2/\[Rho]^2) - (\[Gamma]^2 (-1 + \[Rho]^2) \ \[Sigma]Y Log[\[Gamma]/\[Alpha]])/\[Rho]^2)/(-1 + \[Gamma]^2) 

Simplify[Solve[\[CapitalDelta]\[Sigma]el2 - \[CapitalDelta]\[Sigma]el3 == 0 && \[Alpha] <= \[Gamma] <= 1, \[Gamma]], Assumptions -> {pe > 0, b > 0, P > 0, \[Sigma]Y > 0, \[Alpha] > 0}]

• Is this about Wolfram Mathematica or a purely math question? – Kuba Aug 7 '19 at 9:44
• Both. I've tried to plot the first two expression (fixing certain magnitude of the others parameters). Found the greatest (say $\Delta \sigma_2$) I' ve tried to Solve[$\Delta \sigma_2 - \Delta \sigma_3 == 0 , \gamma$], but it doesn't work. Neither using Simplify and some Assumptions for the parameters. – koda Aug 7 '19 at 9:48
• Then please provide a valid code to work with. – Kuba Aug 7 '19 at 9:50
• Sorry, I'm new here. Hope this works: – koda Aug 7 '19 at 9:59
• @koda Try to edit your answer. To show your code: "To create code blocks or other preformatted text, indent by four spaces or surround with groups of backticks" – Ulrich Neumann Aug 7 '19 at 10:04