What is the smallest possible value of $\sum_{i=1}^n \frac{1}{d_i + 1}$, if the di ’s are constrained to be nonnegative real numbers and satisfy $\sum_{i=1}^n d_i = N$.
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3Welcome to [math.se] SE. Take a [tour]. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an [edit]): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. – Martin R May 01 '22 at 10:59
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2Check this: https://math.stackexchange.com/q/2080451/42969 – it is an immediate application of the inequality between harmonic and arithmetic mean. – Martin R May 01 '22 at 11:02
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By the Cauchy–Schwarz inequality : $$ \left( \sum_{i=1}^n (d_i+1) \right) \left (\sum_{i=1}^n \frac 1 {d_i+1} \right) \geq n^2=\left(\sum_{i=1}^n\sqrt{d_i+1}\frac1{\sqrt{d_i+1}}\right)^2$$
From here you can see that the smallest value is $\frac {n^2}{n+N}$
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3This question seems not to meet the standards for the site. Instead of answering it, why not look for a good duplicate target, or help the user by posting comments suggesting improvements? Please also read the meta announcement regarding quality standards. – Martin R May 01 '22 at 11:01