From the Cauchy-Schwarz inequality deduce that for every function u u u [ a , b ] [a, b] [ a , b ] h > 0 h > 0 h > 0 [ a , b ] [a, b] [ a , b ]
∣ u ( b ) 2 − u ( a ) 2 ∣ ≤ 2 ( ∫ a b u ′ 2 ( t ) h ( t ) d t ) 1 / 2 ( ∫ a b u 2 ( t ) h ( t ) d t ) 1 / 2
|u(b)^2 - u(a)^2| \leq 2 \left( \int_a^b {u'}^2(t) h(t) dt \right)^{1/2} \left( \int_a^b \frac{u^2(t)}{h(t)} dt \right)^{1/2}
∣ u ( b ) 2 − u ( a ) 2 ∣ ≤ 2 ( ∫ a b u ′ 2 ( t ) h ( t ) d t ) 1/2 ( ∫ a b h ( t ) u 2 ( t ) d t ) 1/2 if the two integrals on the right-hand side converge.
By taking a = 0 , b = π / 2 , h ( t ) = 1 a = 0,\ b = \pi/2,\ h(t) = 1 a = 0 , b = π /2 , h ( t ) = 1
u ( t ) = c 1 cos t + c 2 cos 3 t + ⋯ + c n cos ( 2 n − 1 ) t ,
u(t) = c_1 \cos t + c_2 \cos 3t + \cdots + c_n \cos(2n-1)t,
u ( t ) = c 1 cos t + c 2 cos 3 t + ⋯ + c n cos ( 2 n − 1 ) t , where
c j ≥ 0 c_j \geq 0 c j ≥ 0 for every
j j j , deduce the
Carlson inequality ∑ j c j ≤ π ( ∑ j c j 2 ) 1 / 4 ( ∑ j ( j − 1 2 ) 2 c j 2 ) 1 / 4 .
\sum_j c_j \leq \sqrt{\pi} \left( \sum_j c_j^2 \right)^{1/4} \left( \sum_j \left( j - \frac{1}{2} \right)^2 c_j^2 \right)^{1/4}.
j ∑ c j ≤ π ( j ∑ c j 2 ) 1/4 ( j ∑ ( j − 2 1 ) 2 c j 2 ) 1/4 .