55.5. THE PICARD THEOREMS 1757

Therefore, for z ∈ ∂B(0,θR) ,

|H (z)| ≤ |H (0)|+24C ln(

11−θ

). (55.5.26)

By the maximum modulus theorem, the above inequality holds for all |z|< θR also.Next I will use 55.5.23 to get an inequality for | f (z)| in terms of |H (z)|. From 55.5.23,

H (z) = log

(√log( f (z))

2πi−√

log( f (z))2πi

−1

)and so

2H (z) = log

(√log( f (z))

2πi−√

log( f (z))2πi

−1

)2

−2H (z) = log

(√log( f (z))

2πi−√

log( f (z))2πi

−1

)−2

= log

(√log( f (z))

2πi+

√log( f (z))

2πi−1

)2

Therefore, (√log( f (z))

2πi+

√log( f (z))

2πi−1

)2

+

(√log( f (z))

2πi−√

log( f (z))2πi

−1

)2

= exp(2H (z))+ exp(−2H (z))

and (log( f (z))

πi−1)=

12(exp(2H (z))+ exp(−2H (z))) .

Thuslog( f (z)) = πi+

πi2(exp(2H (z))+ exp(−2H (z)))

which shows

| f (z)| =

∣∣∣∣exp[

πi2(exp(2H (z))+ exp(−2H (z)))

]∣∣∣∣≤ exp

∣∣∣∣πi2(exp(2H (z))+ exp(−2H (z)))

∣∣∣∣≤ exp

∣∣∣π2(|exp(2H (z))|+ |exp(−2H (z))|)

∣∣∣≤ exp

∣∣∣π2(exp(2 |H (z)|)+ exp(|−2H (z)|))

∣∣∣= exp(π exp2 |H (z)|) .