Nesbitt's inequality
In mathematics, Nesbitt's inequality states that for positive real numbers "a", "b" and "c", It is an elementary special case (N = 3) of the difficult and much studied Shapiro inequality, and was published at least 50 years earlier. There is no corresponding upper bound as any of the 3 fractions in the inequality can be made arbitrarily large. Proof. First proof: AM-HM inequality. By the AM-HM inequality on formula_2, Clearing denominators yields from which we obtain by expanding the product and collecting like denominators. This then simplifies directly to the final result. Second  proof: Rearrangement. Suppose formula_6, we have that define The scalar product of the two sequences is maximum because of the rearrangement inequality if they are arranged the same way, call formula_10 and formula_11 the vector formula_12 shifted by one and by two, we have: Addition yields our desired Nesbitt's inequality. Third proof: Sum of Squares. The following identity is true for all formula_15 This clearly proves that the left side is no less than formula_17 for positive a, b and c. Note: every rational inequality can be demonstrated by transforming it to the appropriate sum-of-squares identity, see  Hilbert's seventeenth problem. Fourth proof: Cauchy–Schwarz. Invoking the Cauchy–Schwarz inequality on the vectors formula_18 yields which can be transformed into the final result as we did in . Fifth proof: AM-GM. Let formula_20. We then apply the AM-GM inequality to obtain the following because formula_22 Substituting out the formula_23 in favor of formula_24 yields which then simplifies to the final result. Sixth proof: Titu's lemma. Titu's lemma, a direct consequence of the Cauchy–Schwarz inequality, states that for any sequence of formula_27 real numbers formula_28 and any sequence of formula_27 positive numbers formula_30, formula_31 . We use the lemma on formula_32 and formula_33. This gives, This results in, Seventh proof: Using homogeneity. As the left side of the inequality is homogeneous, we may assume formula_37. Now define formula_38, formula_39, and formula_40. The desired inequality turns into formula_41, or, equivalently, formula_42. This is clearly true by Titu's Lemma. Eighth proof: Jensen inequality. Define formula_43 and consider the function formula_44. This function can be shown to be convex in formula_45 and, invoking Jensen inequality, we get A straightforward computation yields Ninth proof: Reduction to a two-variable inequality. By clearing denominators, It now suffices to prove  that formula_49 for formula_50, as summing this three times for formula_51 and formula_52 completes the proof. As formula_53 we are done.