An extended Jordan’s inequality in exponential type

In this paper, we establish the following double exponential type Jordan’s inequality ( sin r r ) p + α ( r 2 − x 2 ) ≤ ( sin x x ) p ≤ ( sin r r ) p + β ( r 2 − x 2 ) for x ∈ ( 0 , r ] , 0 < r ≤ π / 2 , and p > 0 .