p-harmonic functions by way of intrinsic mean value properties

Lemma A.1.

Let T ∈ [ 0 , π 2 ] and γ > 0 . Then

whenever | t | ⩽ T .

Proof.

Let a ∈ [ 0 , 1 ] and define φ ± : [ a , 1 ] → ℝ by

Direct computation shows that

Therefore, φ ± is positive and increasing in [ a , 1 ] . In particular, φ ± ⁢ ( x ) ⩽ φ ± ⁢ ( 1 ) for every x ∈ [ a , 1 ] . Then the result follows by letting a = cos ⁡ T and performing the change of variables x = cos ⁡ t . ∎

Lemma A.2.

Let γ ∈ ( 0 , 1 ) . Then

for all x ∈ ( 0 , 1 ] .

Proof.

Let us recall the Taylor series of f ⁢ ( x ) = ( 1 + x ) γ :

for | x | ⩽ 1 , where

for each k ∈ ℕ . Observe that, since γ ∈ ( 0 , 1 ) , we have that

We can rewrite the left-hand side in (A.1) by replacing (A.2):

Hence, (A.1) follows from the fact that

for every x ∈ ( 0 , 1 ) . In fact, every coefficient in the above series is nonpositive, that is,

for every k ∈ ℕ . ∎