Abstract
In this paper we investigate properties of the Steiner symmetrization in the complex plane. We use two recursive dynamic processes in order to derive some sharp inequalities on analytic functions in the unit disk. We answer a question that was asked by Albert Baernstein II, regarding the coefficients of circular symmetrization. We mostly deal with the Steiner symmetrization $G$ of an analytic function $f$ in the unit disk $U$. We pose few problems we can not solve. An intriguing one is that of t
Results & Lemmas (29)
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Proposition 1.3.
Proposition 1.3. If 2 ≤p ≤∞, 0 < α < ∞, then there exists a function f ∈S(p, α) such that: N(p, α) = Z 2π 0 |f ′(eiθ)|dθ.
Proposition 1.3. If 2 ≤p ≤∞, 0 < α < ∞, then there exists a function f ∈S(p, α) such that: N(p, α) = Z 2π 0 |f ′(eiθ)|dθ.
Proposition 1.6.
Proposition 1.6. If 2 ≤p ≤∞, 0 < α < ∞and if f ∈S(p, α) was extremal for N(p, α), then the simply connected domain f(U) can have no slits.
Proposition 1.6. If 2 ≤p ≤∞, 0 < α < ∞and if f ∈S(p, α) was extremal for N(p, α), then the simply connected domain f(U) can have no slits.
Proposition 1.7.
Proposition 1.7. If 2 ≤p ≤∞, 0 < α < ∞and if f ∈S(p, α) was extremal for N(p, α), then the simply connected domain f(U) is a convex domain.
Proposition 1.7. If 2 ≤p ≤∞, 0 < α < ∞and if f ∈S(p, α) was extremal for N(p, α), then the simply connected domain f(U) is a convex domain.
Proposition 1.8.
Proposition 1.8. If 2 ≤p ≤∞, 0 < α < ∞and if f ∈S(p, α) was extremal for N(p, α), and if |a| < 1 then either: |f ′(a)| ≤ 1 1 −|a|2, 3
Proposition 1.8. If 2 ≤p ≤∞, 0 < α < ∞and if f ∈S(p, α) was extremal for N(p, α), and if |a| < 1 then either: |f ′(a)| ≤ 1 1 −|a|2 , 3
Theorem 6.
Theorem 6. ([1]) If Φ is a convex non-decreasing function on (−∞, ∞), f ∈H(U) and F as above, then for all 0 ≤r < 1 we have: Z π −π Φ(log…
Theorem 6. ([1]) If Φ is a convex non-decreasing function on (−∞, ∞), f ∈H(U) and F as above, then for all 0 ≤r < 1 we have: Z π −π Φ(log |f(reiθ)|)dθ ≤ Z π −π Φ(log |F(reiθ)|)dθ. If we choose in Theorem 6 above, Φ(x) = e2x and assume that we have the following expansions: f(z) = P∞ n=0 anzn and F(z) = P∞ n=0 Anzn, then we obtain the inequality P∞ n=0 |an|2r2n ≤P∞ n=0 |An|2r2n for 0 ≤r < 1. By
Theorem 2.2.
Theorem 2.2. If f(z) = P∞ n=0 anzn is analytic, one-to-one in U and f(U) has a finite area. If F(z) = P∞ n=0 Anzn is the circular…
Theorem 2.2. If f(z) = P∞ n=0 anzn is analytic, one-to-one in U and f(U) has a finite area. If F(z) = P∞ n=0 Anzn is the circular symmetrization of 6
Theorem 2.2
Theorem 2.2 answers the problem mentioned above that was raised by Albert Baernstein II. The answer in negative. Next, let f(z) = P∞ n=0…
Theorem 2.2 answers the problem mentioned above that was raised by Albert Baernstein II. The answer in negative. Next, let f(z) = P∞ n=0 anzn, z ∈U, be analytic and one-to-one, and assume that ∂D = ∂f(U) is rectifiable.
Theorem 2.5.
Theorem 2.5. If f ∈H(U), f is one-to-one, and the boundary curve f(eiθ) | 0 ≤θ < 2π is rectifiable. If F is the circular symmetrization…
Theorem 2.5. If f ∈H(U), f is one-to-one, and the boundary curve {f(eiθ) | 0 ≤θ < 2π} is rectifiable. If F is the circular symmetrization (P¨olya) of f and if G is the Steiner symmetrization of f (F(0) = G(0) = |f(0)|), then the boundary curves {F(eiθ) | 0 ≤θ < 2π} and {G(eiθ) | 0 ≤ θ < 2π} are rectifiable, and we have the following two inequalities: Z 2π 0 |G′(eiθ)|dθ ≤ Z 2π 0 |f ′(eiθ)|dθ, Z 2π 0 |F ′(eiθ)|dθ ≤
Theorem 2.7.
Theorem 2.7. If f is analytic and one-to-one in U and if G is the Steiner symmetrization of f, then for any r, 0 ≤r < 1 we have the…
Theorem 2.7. If f is analytic and one-to-one in U and if G is the Steiner symmetrization of f, then for any r, 0 ≤r < 1 we have the inequality: Z 2π 0 |f(reiθ)|2dθ ≤ Z 2π 0 |G(reiθ)|2dθ.
Proposition 2.9.
Proposition 2.9. If 2 ≤p ≤∞, 0 < α < ∞, and if f ∈S(p, α) is an extremal function for N(p, α) then we may assume that the domain f(U) is…
Proposition 2.9. If 2 ≤p ≤∞, 0 < α < ∞, and if f ∈S(p, α) is an extremal function for N(p, α) then we may assume that the domain f(U) is circular symmetric (P¨olya symmetric).
Theorem 3.4.
Theorem 3.4. If 0 < α < ∞, and if f ∈S(2, α) and also φn ∞ n=1 is any sequence of real numbers, then we have the double inequality: 0 < ∞ Y…
Theorem 3.4. If 0 < α < ∞, and if f ∈S(2, α) and also {φn}∞ n=1 is any sequence of real numbers, then we have the double inequality: 0 < ∞ Y n=1 cφn ≤1.
Theorem 3.5.
Theorem 3.5. Let D be a bounded domain that contains the origin, 0. Then, there exists a disk B whose center is the origin, 0 and there…
Theorem 3.5. Let D be a bounded domain that contains the origin, 0. Then, there exists a disk B whose center is the origin, 0 and there exists a sequence of φn-deformations of D that will be denoted by {Dn}∞ n=1 so that Dn →B, (The assumption means that D1 is the φ1-deformation of D, and Dn+1 is the φn+1-deformation of Dn). Moreover, the disk B is unique in the sense that if {D′ n}∞ n=1 is the sequence of φ′ n-deformations of D that satisfies D′ n →B′, then d(B) = d(B′)
Theorem 3.6.
Theorem 3.6. If 0 < α < ∞, and if G(z) = max 1, α z, then G ∈S(2, α) and we have: N(2, α) = Z 2π 0 |G′(eiθ)|dθ = 2π max 1, α.
Theorem 3.6. If 0 < α < ∞, and if G(z) = max{1, α}z, then G ∈S(2, α) and we have: N(2, α) = Z 2π 0 |G′(eiθ)|dθ = 2π max{1, α}.
Theorem 3.8.
Theorem 3.8. Let f ∈H(U), f(0) = 0. Then for each 0 ≤r < 1 we have the following inequality: 1 2π Z 2π 0 |f(reiθ)|2dθ 1/2 ≤r 2π Z 2π 0…
Theorem 3.8. Let f ∈H(U), f(0) = 0. Then for each 0 ≤r < 1 we have the following inequality: 1 2π Z 2π 0 |f(reiθ)|2dθ 1/2 ≤r 2π Z 2π 0 |f ′(reiθ)|dθ. In particular we have ||f||2 ≤||f ′||1. Both inequalities above are sharp.
Lemma 4.2.
Lemma 4.2. Let the function f ∈H(U) satisfy f(0) = 0. Then for any value of r, 0 ≤r < 1 we have the following estimate: max 0≤θ<2π…
Lemma 4.2. Let the function f ∈H(U) satisfy f(0) = 0. Then for any value of r, 0 ≤r < 1 we have the following estimate: max 0≤θ<2π |f(reiθ)| ≤r 2 Z 2π 0 |f ′(reiθ)|dθ. In particular we have the inequality ||f||∞≤π||f ′||1. Both inequalities above are sharp.
Theorem 4.3.
Theorem 4.3. Let f ∈H(U) satisfy f(0) = 0, and let 2 ≤p ≤∞. Then for each value of r, 0 ≤r < 1 we have: 1 2π Z 2π 0 |f(reiθ)|pdθ 1/p ≤ r…
Theorem 4.3. Let f ∈H(U) satisfy f(0) = 0, and let 2 ≤p ≤∞. Then for each value of r, 0 ≤r < 1 we have: 1 2π Z 2π 0 |f(reiθ)|pdθ 1/p ≤ r 2π2/p Z 2π 0 |f ′(reiθ)|dθ. In particular also the following inequality holds true: ||f||p ≤π(p−2)/p||f ′||1.
Theorem 4.4.
Theorem 4.4. If f ∈H(U) and if ||f ′||1 < ∞, then f ∈Hp(U) for all p, 2 ≤p ≤∞and the following inequality holds true: ||f||p ≤π(p−2)/p||f…
Theorem 4.4. If f ∈H(U) and if ||f ′||1 < ∞, then f ∈Hp(U) for all p, 2 ≤p ≤∞and the following inequality holds true: ||f||p ≤π(p−2)/p||f ′||1 + |f(0)|. 17
Proposition 6.1.
Proposition 6.1. 1. exp(D∗) is the circular symmetrization of exp(D), where D is a domain and D∗is the Steiner symmetrization of this…
Proposition 6.1. 1. exp(D∗) is the circular symmetrization of exp(D), where D is a domain and D∗is the Steiner symmetrization of this domain. 2. If for all a ∈R, l(a) < 2π, then the intersection arcs {eaeiθ | |θ| < (1/2)l(a)} are simple (i.e. they do not pass through any point more than once). 3. If for all a ∈R, l(a) < 2π, then exp(D∗) is a simply connected domain.
Theorem 6.2.
Theorem 6.2. Let f ∈H(U) and let us denote D = f(U), and assume that for any a ∈R we have l(a) < 2π. Also suppose that f(0) ≥0. Let F ∈H(U)…
Theorem 6.2. Let f ∈H(U) and let us denote D = f(U), and assume that for any a ∈R we have l(a) < 2π. Also suppose that f(0) ≥0. Let F ∈H(U) be a conformal mapping of U onto D∗, where F(0) = |f(0)| = f(0) and where D∗is the Steiner symmetrization of D. If Φ is a convex non-decreasing function on (−∞, ∞), then for all r, 0 ≤ r < 1, we have: Z π −π Φ(ℜ{f(reiθ)})dθ ≤ Z π −π Φ(ℜ{F(reiθ)})dθ. 20
Proposition 6.1
Proposition 6.1(3). Also we have: exp(F(0)) = exp(|f(0)|) = exp(f(0)), where the last equality follows by our assumption, f(0) ≥0. To sum…
Proposition 6.1(3). Also we have: exp(F(0)) = exp(|f(0)|) = exp(f(0)), where the last equality follows by our assumption, f(0) ≥0. To sum up we have g(z) = exp(f(z)) ∈H(U) where by the above no- tations: g : U →exp(D) = g(U). The mapping G(z) = exp(F(z)) is a conformal and onto mapping G : U →exp(D∗) that satisfies G(0) = exp(F(0)) = exp(f(0)) = g(0) = |g(0)|. The simply connected domain exp(D∗) is the circular symmetrization of exp(D) = g(U). Thus the pair of mappings g, G satisfy all the assump
Theorem 6
Theorem 6 in [1]. Using this theorem we obtain: Z π −π Φ(log |g(reiθ)|)dθ ≤ Z π −π Φ(log |G(reiθ)|)dθ, for any convex and non-decreasing Φ…
Theorem 6 in [1]. Using this theorem we obtain: Z π −π Φ(log |g(reiθ)|)dθ ≤ Z π −π Φ(log |G(reiθ)|)dθ, for any convex and non-decreasing Φ on (−∞, ∞), and any r, 0 ≤r < 1. Plugging in the expressions g(z) = exp(f(z)) and G(z) = exp(F(z)) we obtain: Z π −π Φ(log | exp(f(reiθ))|)dθ ≤ Z π −π
Corollary 6.3.
Corollary 6.3. Let f ∈H(U) satisfy l(a) < 2π for all a ∈R and f(0) ≥0, and let F: U →f(U)∗(f(U)∗is the Steiner symmetrization of f(U)) be a…
Corollary 6.3. Let f ∈H(U) satisfy l(a) < 2π for all a ∈R and f(0) ≥0, and let F : U →f(U)∗(f(U)∗is the Steiner symmetrization of f(U)) be a conformal onto with F(0) = f(0), then: 1. For any 0 < p we have: Z π −π exp pℜ{f(reiθ)} dθ ≤ Z π −π exp
Corollary 6.5.
Corollary 6.5. Let f ∈H(U) satisfy l(a) < 2π for all a ∈R, and f(0) ≥0, and let F: U →f(U)∗be conformal, onto with F(0) = f(0), then for…
Corollary 6.5. Let f ∈H(U) satisfy l(a) < 2π for all a ∈R, and f(0) ≥0, and let F : U →f(U)∗be conformal, onto with F(0) = f(0), then for any p > 1 we have the following inequality: Z π −π exp ℜ{f(reiθ)} p + dθ ≤ Z π −π exp
Proposition 7.1.
Proposition 7.1. ∞ X n=1 log 1 cφn < ∞. (7.7) ∞ X n=1 (1 −cφn) < ∞. (7.8)
Proposition 7.1. ∞ X n=1 log 1 cφn < ∞. (7.7) ∞ X n=1 (1 −cφn) < ∞. (7.8)
Proposition 7.2.
Proposition 7.2. If 0 < α < ∞, and if f ∈S(2, α), and if φn ∞ n=1 is any sequence of real numbers, then the infinite product: B φn (z) = ∞ Y…
Proposition 7.2. If 0 < α < ∞, and if f ∈S(2, α), and if {φn}∞ n=1 is any sequence of real numbers, then the infinite product: B{φn}(z) = ∞ Y n=1 z −cφn 1 −cφnz , is a Blaschke product, i.e. it is uniformly convergent on compact subsets of U, and {cφn}∞ n=1 is the zero set of the resulting bounded (by 1) analytic function, B{φn}(z).
Proposition 7.4.
Proposition 7.4. ∞ X n=1 log min g ′ φn(0), 1 α · ||gφn||2 < ∞. (7.9)
Proposition 7.4. ∞ X n=1 log min g ′ φn(0), 1 α · ||gφn||2 < ∞. (7.9)
Theorem 9.1.
Theorem 9.1. Let f(z) ∈S(2, α), then for any sequence φn ∞ n=1 of real numbers the limit function F = limn→∞(... ((gφ1)φ2)...)φn exists and…
Theorem 9.1. Let f(z) ∈S(2, α), then for any sequence {φn}∞ n=1 of real numbers the limit function F = limn→∞(. . . ((gφ1)φ2) . . .)φn exists and the convergence is uniform on compact subsets of U. The image F(U) is a Steiner symmetric domain that includes z = 0 and F ∈S(2, α).
Theorem 9.2.
Theorem 9.2. 1. Let f ∈S(2, α) and let φn ∞ n=1 be any sequence of real numbers. Let cφn ∞ n=1 be the corresponding sequence of the…
Theorem 9.2. 1. Let f ∈S(2, α) and let {φn}∞ n=1 be any sequence of real numbers. Let {cφn}∞ n=1 be the corresponding sequence of the shrinking factors. Then we have the following estimate: max {1, α} · 1 π Z 2π 0 ℜ{f(eiθ}ℜ{eiθf ′(eiθ)}dθ −1/2 ≤
Corollary 9.4.
Corollary 9.4. Let f ∈S(2, α) and let φn ∞ n=1 be any sequence of real numbers. Let G be the limiting function of the newer recursive…
Corollary 9.4. Let f ∈S(2, α) and let {φn}∞ n=1 be any sequence of real numbers. Let G be the limiting function of the newer recursive process, i.e. G = limn→∞(. . . ((fφ1)φ2) . . .)φn. Then we have the sharp estimate: max {1, α}· 1 π Z 2π 0 ℜ{f(eiθ)}ℜ{eiθf ′(eiθ)}dθ −1/2 ≤max 1