A treatise on the theory of Bessel functions, by G. N. Watson.

13-74] INFINITE INTEGRALS 447
We twice integrate by parts the last portion of the second term in the
integral thus
2v2 sinh t IK' (2x sinh t) cosh 2vt dt
= v sinh t sinh 2vtKo' (2x sinh t)j -v - {sinh tKo' (2x sinh t)} sinh 2ivtdt
- - | j d {sinh tK' (2x sinh t)} sinh 2tcdt
00
-= - 2x sinh t cosh tKo (2x sinh t) sinh 2ztdt
- o
-= [- x sinh t cosh tIfo (2w sinh t) cosh 2vtl
f00 rf
+ o d [sinh t cosh t K (2x sinh t)] cosh 2vt dt
f i[x cosh 2tKo (2x sinh t) +. 2x sinh t cosh2 tK,' (2x sinh t)] cosh 2vt dt;
the simplification after the second step is produced by using the differential
equation
zKo" (z) + It' (z)-zKO (z). = 0.
The integral under discussion consequently reduces to
f [- 2x sinh2 t Ko (2x sinh t) - 2x sinh3 tKo' (2x sinht)] cosh 2vt dt.o
_x sinh8 t
L= [- e osh — K0 (2x sinh t) cosh 2 Pt]
|_ cosh t v Jo
+ x Ko (2x sinh t) 2 sinh cosh 2vt t + osh t cosh 2vt dt
w= s JT (2x sinh t) [tanh2 t cosh 2 t + 2v sinh3 t sech t sinh 2vt] dt,
and this is positive because the integrand is positive; hence the differential
coefficient of
(2 2- v2) {J2 () + Yv2 (X)
is positive, and the result is established.
Since the limits of both the functions
X {fJ2 (x) + Y 2 (X)j, (x2 - 2)i {J,2 (X) + Y2 (x)}
are 2/v7, it follows from the last two results that when x > v>,
(1) (: ) > J^ (X) + Y2 () > 2/
An elementary proof of the last inequality (with various related inequalities) was
deduced by Schafheitlin, Berliner Sitzungsberichte, v. (1906), p. 86, from the formula
(cf. ~ 5-14)
(42 - 1) 2 (t) dt
4 ( 2)_2
where Vv (x) -= aJd (x) + b J, (x).