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@CarloBeenakker does that mean the integration is not possible in close from? Can you give me a hint to do that. Yes I posted this in overflow as no one answered in MSE Thank you.
Thank you for you reply, Its really helpful but the last line I did not get yet. Did you mean sum of logconvex is logconvex for any positive $\epsilon_i$? But in my case the sum is inside the log function. And can you give me any ref. regarding last line that this is applicable for my case where the sum is inside log function. Again thank you very much.