An extension of the Geometric Modulus Principle to holomorphic and harmonic functions
AAn extension of the Geometric Modulus Principle toholomorphic and harmonic functions
Matt Hohertz
Department of Mathematics, Rutgers [email protected] 23, 2021
Abstract
Kalantari’s Geometric Modulus Principle describes the local behavior of the modulusof a polynomial. Specifically, if p ( z ) = a + n ∑ j = k a j ( z − z ) j , a a k a n (cid:54) =
0, then the complexplane near z = z comprises 2 k sectors of angle π k , alternating between arguments of ascent (angles θ where (cid:12)(cid:12) p ( z + te i θ ) (cid:12)(cid:12) > | p ( z ) | for small t ) and arguments of descent (where theopposite inequality holds). In this paper, we generalize the Geometric Modulus Principleto holomorphic and harmonic functions. As in Kalantari’s original paper, we use theseextensions to give succinct, elegant new proofs of some classical theorems from analysis. Introduction
For a holomorphic function p and fixed z ∈ C , we define an argument of ascent of p at z tobe a value θ ∈ [ , π ) for which there exists t ∗ > (cid:12)(cid:12)(cid:12) p ( z + te i θ ) (cid:12)(cid:12)(cid:12) > | p ( z ) | (1)for all t ∈ ( , t ∗ ) ; we define arguments of descent analogously. We define the cone of ascentC a ( p , z ) to be the intersection of the set of arguments of ascent with [ , π ) , and define the cone of descent C d ( p , z ) analogously.Kalantari’s Geometric Modulus Principle is the following. Theorem I (Geometric Modulus Principle)
Let p ( z ) be a nonconstant polynomial. If p ( z ) = , then C a ( p , z ) = [ , π ) . If p ( z ) (cid:54) = , then C a ( p , z ) and C d ( p , z ) partition the unit circleinto alternating sectors of angle π k . Specifically, if re i α is the polar form of p ( z ) p ( k ) ( z ) then θ is an argument of ascent if − α − π k < θ < − α + π k (cid:18) mod π k (cid:19) . (2)1 a r X i v : . [ m a t h . C V ] F e b he interiors of the remaining sectors are arguments of descent. A boundary line between twosectors is neither an argument of ascent nor an argument of descent . Except in its last sentence, Theorem I is a straightforward corollary of the following lemma.
Lemma 1
Let p ( z ) be a polynomial of degree n ≥ . Suppose that z is a complex num-ber, p ( z ) (cid:54) = , and k : = min (cid:96) ≥ { (cid:96) : p ( (cid:96) ) ( z ) (cid:54) = } . Given a real number θ , define F θ ( t ) : = (cid:12)(cid:12) p (cid:0) z + te i θ (cid:1)(cid:12)(cid:12) . Finally, let re i α be the polar form of p ( z ) p ( k ) ( z ) . ThenF ( k ) θ ( ) = r cos ( α + k θ ) . P ROOF
We paraphrase the proof from Kalantari [2011]. First, by Taylor’s Theorem we assumewithout loss of generality that z =
0. Then F θ ( t ) = p ( te i θ ) · p ( te i θ ) (3) = (cid:32) a + ∑ j ≥ k a j t j e i j θ (cid:33) · (cid:32) a + ∑ j ≥ k a j t j e − i j θ (cid:33) (4) = | a | + t k · (cid:16) a a k e ik θ + a a k e − ik θ (cid:17) + t k + · ( · · · ) . (5)The terms of this Maclaurin series correspond to the derivatives of F θ ( t ) at t =
0, so that F ( k ) θ ( ) = k ! · (cid:16) a a k e ik θ + a a k e − ik θ (cid:17) . (6)Continuing, we obtain F ( k ) θ ( ) = · Re (cid:16) p ( ) p ( k ) ( ) e ik θ (cid:17) (7) = · Re (cid:16) re i ( α + k θ ) (cid:17) (8) = r cos ( α + k θ ) . (9) (cid:4) Thus the proof of Theorem I is very simple.P
ROOF (T HEOREM
I) At θ such that F ( k ) θ ( ) (cid:54) =
0, the behavior of (cid:12)(cid:12) p ( z + te i θ ) (cid:12)(cid:12) for smallpositive t follows directly from Lemma 1. On the other hand, suppose that some θ such that F ( k ) θ ( ) = { z : | p ( z ) | > | p ( z ) |} under p would contain some of its own boundary points. But, by the continuity of p , this preimagemust be open. The case in which θ is a direction of descent is analogous. (cid:4) Note that the original theorem here reads “either ... or ... descent.” Yuefei Wang, personal communication, February 19, 2021 Extension to holomorphic functions
Lemma 2
In Lemma 1, the requirement that p ( z ) be a non-constant polynomial may be relaxedto a requirement that p ( z ) be non-constant and holomorphic near z . P ROOF
The proof of the lemma requires only that p ( z ) be complex analytic (equivalently,holomorphic) with at least one non-constant term. In particular:(i) By Mertens’ Theorem, Equation (5) continues to hold as long as one of the two seriesconverges absolutely. But holomorphic functions converge absolutely within their discsof convergence.(ii) Equation (6) continues to hold because the series of Equation (5) can be differentiatedterm-by-term in its interval of convergence [Apostol, 1952, Theorem 3]. (cid:4) Since holomorphic functions are continuous, we can extend the Geometric Modulus Prin-ciple to all holomorphic functions.
Theorem II (Extended Geometric Modulus Principle)
In Theorem I, the requirement thatp ( z ) be a non-constant polynomial may be relaxed to a requirement that p ( z ) be non-constantand holomorphic near z . Example
Define ω : = e i π and f ( z ) : = − ω z . Near z = f ( z ) has the expansion f ( z ) = + ω z + iz + · · · , (10)so that re i α = f ( ) · f ( ) ( ) (11) = ω , (12)yielding α = π . Thus we expect that C a ( f , ) = (cid:16) − π , π (cid:17) ∪ (cid:18) π , π (cid:19) ∪ (cid:18) π , π (cid:19) ( mod 2 π ) . (13)Figure 1, which graphs the lemniscate {| f ( z ) | < | f ( ) |} , bears this prediction out.Our extension of the Geometric Modulus Principle has, as a direct corollary, the general Maximum Modulus Principle. From here on, we will simply refer to Theorem II as the Geometric Modulus Principle without clarifying thatit is an extension. cf . [Kalantari, 2011, Theorem 2] (cid:12)(cid:12)(cid:12) − e i π / z (cid:12)(cid:12)(cid:12) < Theorem III (Maximum Modulus Principle)
Let Ω be a connected, open subset of C ; letf : Ω → C be holomorphic. If | f ( z ) | ≥ | f ( z ) | for all z in a neighborhood of some z ∈ Ω , thenf is constant on Ω . P ROOF
By the Geometric Modulus Principle, if f is non-constant then it has at least one di-rection of ascent at each z ∈ Ω ; in particular, | f | cannot attain a local maximum in Ω . (cid:4) In this section we extend the Geometric Modulus Principle to harmonic functions, using thefact that all harmonic functions are locally the real part of some holomorphic function. Definean argument of ascent (resp., argument of descent ) of a harmonic function u at z to be an angle θ such that u (cid:0) z + te i θ (cid:1) > u ( z ) [resp., u (cid:0) z + te i θ (cid:1) < u ( z ) ] for sufficiently small positive t .As before, let C a ( u , z ) and C d ( u , z ) represent the cones of, respectively, ascent and descent of u at z .We first observe that Re ( f ) and e f have the same cones of ascent. In what follows, assume Ω to be an open, connected subset of C . Lemma 3
Let u : Ω → R be the real part of a holomorphic function f : Ω → C . Then forz ∈ Ω , C a ( u , z ) = C a (cid:0) e f , z (cid:1) and C d ( u , z ) = C d (cid:0) e f , z (cid:1) . ROOF
Note that (cid:12)(cid:12)(cid:12) e f ( z ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) e u ( z )+ iv ( z ) (cid:12)(cid:12)(cid:12) (14) = (cid:12)(cid:12)(cid:12) e u ( z ) (cid:12)(cid:12)(cid:12) (15) = e u ( z ) , (16)so that (cid:12)(cid:12)(cid:12) e f ( z ) (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) e f ( z ) (cid:12)(cid:12)(cid:12) and u ( z ) − u ( z ) have the same sign for all z , z ∈ C . (cid:4) The next three lemmas relate C a ( f , z ) and C a ( e f , z ) . Lemma 4
If k ≥ and f ( z ) is k times differentiable, then [ exp ( f ( z ))] ( k ) = k − ∑ j = (cid:18) k − j (cid:19) f ( k − j ) ( z ) · [ exp ( f ( z ))] ( j ) . (17)P ROOF
For k =
1, we simply apply the Chain Rule. For k >
1, we may apply the GeneralLeibniz Rule to [ exp ( f ( z ))] (cid:48) : [ exp ( f ( z ))] ( k ) = (cid:2) f (cid:48) ( z ) · exp ( f ( z )) (cid:3) ( k − ) (18) = k − ∑ j = (cid:18) k − j (cid:19) (cid:2) f (cid:48) ( z ) (cid:3) ( k − − j ) [ exp ( f ( z ))] ( j ) (19) = k − ∑ j = (cid:18) k − j (cid:19) f ( k − j ) ( z ) · [ exp ( f ( z ))] ( j ) . (20) (cid:4) Lemma 5
Let f : Ω → C be holomorphic at z ∈ Ω and define g ( z ) : = exp ( f ( z )) . Supposethat, for some k ≥ and z ∈ Ω ,(1) f ( k ) ( z ) (cid:54) = and(2) f ( (cid:96) ) ( z ) = for each (cid:96) such that ≤ (cid:96) < k.Then(i) g ( k ) ( z ) = g ( z ) · f ( k ) ( z ) (cid:54) = and(ii) g ( (cid:96) ) ( z ) = for each (cid:96) such that ≤ (cid:96) < k. P ROOF
Both items follow readily from applying the hypotheses to Equation (17). (cid:4)
In what follows, if x ∈ R and S ⊆ R / ( π Z ) then let S + x denote the set S + x : = { y + x : y ∈ S } ( mod 2 π ) . (21)For example, [ π , π ) + π = [ , π ) ∪ [ π , π ) . 5 emma 6 Let f be holomorphic near z ∈ Ω , and define g ( z ) : = exp ( f ( z )) . Moreover, let kretain its meaning from Lemma 5. ThenC a ( g , z ) = − (cid:16) f ( k ) ( z ) (cid:17) − π k , − (cid:16) f ( k ) ( z ) (cid:17) + π k (cid:18) mod π k (cid:19) (22) = C a ( f , z ) − arg ( f ( z )) k ( if f ( z ) (cid:54) = ) . (23) The set C d ( g , z ) equals the interior of the remaining sectors. P ROOF
By Theorem II and Lemma 5, C a ( g , z ) and C d ( g , z ) partition the unit disc into 2 k sectors, which are determined by the following corollary of Lemma 5(i): g ( z ) g ( k ) ( z ) = | g ( z ) | · f ( k ) ( z ) . (24)Thus Equation (22) follows from Equations (2) and (24).Now assume that f ( z ) (cid:54) = re i α be the polar form of f ( z ) f ( k ) ( z ) . Then f ( z ) · g ( z ) g ( k ) ( z ) = | g ( z ) | · re i α (25) g ( z ) g ( k ) ( z ) = | g ( z ) | | f ( z ) | · re i [ α + arg ( f ( z ))] . (26)Equation (23) follows readily. (cid:4) Combining Lemmas 3 and 6, we extend the Geometric Modulus Principle to harmonicfunctions:
Theorem IV (Geometric Modulus Principle for harmonic functions)
Let u : Ω → R be thereal part of a holomorphic function f : Ω → C that satisfies the hypotheses of Lemma 5. ThenC a ( u , z ) and C d ( u , z ) partition the unit circle into k alternating sectors of angle π k . Specifi-cally, θ is an argument of ascent of u if − (cid:16) f ( k ) ( z ) (cid:17) − π k < θ < − (cid:16) f ( k ) ( z ) (cid:17) + π k (cid:18) mod π k (cid:19) . (27) The interiors of the remaining sectors are arguments of descent. A boundary line between twosectors is neither an argument of ascent nor an argument of descent.
With Theorem IV, we prove the Maximum Principle for harmonic functions:
Theorem V
If the harmonic function u : Ω → C attains a local extremum at z ∈ Ω , then u isconstant. P ROOF If u is non-constant, then it is locally the real part of a non-constant holomorphicfunction. Thus, by Theorem IV, it has at least one argument of ascent and at least one argumentof descent. (cid:4) Theorem VI C a ( Im ( f ) , z ) = C a ( Re ( f ) , z ) + π k (28) and C d ( Im ( f ) , z ) = C d ( Re ( f ) , z ) + π k . (29)P ROOF
Define u : = Re ( f ( z )) (30) v : = Im ( f ( z )) , (31)and define h ( z ) = − i · f ( z ) . Now, v = Re ( h ( z )) , so that θ is an argument of ascent of h if − (cid:16) h ( k ) ( z ) (cid:17) − π k < θ < − (cid:16) h ( k ) ( z ) (cid:17) + π k (cid:18) mod 2 π k (cid:19) . (32) − (cid:104) arg (cid:16) f ( k ) ( z ) (cid:17) − π (cid:105) − π k < θ < − (cid:104) arg (cid:16) f ( k ) ( z ) (cid:17) − π (cid:105) + π k (cid:18) mod 2 π k (cid:19) . (33) − arg (cid:16) f ( k ) ( z ) (cid:17) k < θ < − arg (cid:16) f ( k ) ( z ) (cid:17) + π k (cid:18) mod 2 π k (cid:19) . (34) (cid:4) Example
Suppose that f ( z ) = z + ( + i √ ) ; then f , e f , Re ( f ) , and Im ( f ) all partitionthe unit circle into four sectors of angle π . The respective cones of ascent of these functionsare visible in Figure 2. Note that e f and Re ( f ) have the same cones of ascent, as Lemma3 implies; moreover, f [resp., Im ( f ) ] has the same sections rotated anticlockwise by π asLemma 6 implies (resp., by π as Theorem VI implies).Finally, we prove that zeros of non-constant harmonic functions are not isolated, in contrastto the holomorphic case. Theorem VII
If the non-constant harmonic function u : Ω → C has a zero at z = z , then italso has a zero in every punctured disc { z : 0 < | z − z | < t } . P ROOF
By Theorem IV the cones of ascent and descent partition the unit disc into at least twosectors, which are bounded by the level curve { z : | p ( z ) | = } . (cid:4) f ( z ) : = z + ( + √ i ) , the lemniscates | f ( z ) | > | e f ( z ) | > e , Re [ f ( z )] >
1, and Im [ f ( z )] > √ Acknowledgements
I would like to thank Yuefei Wang for his comments, which pointed out an error in an earlydraft.
References [1] Bahman Kalantari. A Geometric Modulus Principle for Polynomials.
The American Math-ematical Monthly , 118(10):931–935, Dec 2011. doi: 10.4169/amer.math.monthly.118.10.931.[2] T. M. Apostol. Term-wise differentiation of power series.