Example 1.8- For any real number 0 and any positive integer n show the inequality 4 Cos 2 + ∞ Σ΄ n=0 cos 20 3 cos no π-0

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Example 1.8- For any real number 0 and any positive integer n show the inequality 4 Cos 2 + ∞ Σ΄ n=0 cos 20 3 cos no π-0 n+1 2 = + m Σ ... n=1 + This was shown by Rogosinski and Szegö (1928). Verblunsky (1945) gave another proof. Koumandos (2001) obtained the lower bound -41/96 for n ≥ 2. Note that cos ne n+1 > - for 0 < 0 < 2л. For the simpler cosine sum Young (1912) showed that cos no > -1 n 1 2 sin cos 0 log 2 sin log (2 sin) Problems and Solutions in Real Analysis for any and positive integer m 2 2. Brown and Koumandos (1997) improved this by replacing -1 by -5/6.