Results & Lemmas (9)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
LEMMA 1.1.
LEMMA 1.1. If D is a bounded domain, the Bergman kernel func- tion KD(z, z) is strictly plurisubharmonic and (1.2) 1/ωφ) ^ KD(z, z) ^…
LEMMA 1.1. If D is a bounded domain, the Bergman kernel func- tion KD(z, z) is strictly plurisubharmonic and (1.2) 1/ωφ) ^ KD(z, z) ^ l/π"(l(z)Y n , where l(z) = min r e 9 2 ) ρ(τ, z), ρ(τ, z) = max,- {| τ5 — zά |, j = 1, , n) and co(D) signifies the euclidean volume of D. 569
LEMMA 2.1.
LEMMA 2.1. For a fixed value t, a holomorphic univalent function w = f(z) of D have convex image Δt of Dt defined by (1.1) if and only if…
LEMMA 2.1. For a fixed value t, a holomorphic univalent function w = f(z) of D have convex image Δt of Dt defined by (1.1) if and only if at every point z on the boundary dDt
Theorem 3.1
Theorem 3.1). Hence, for any holomorphic mapping g(z) of D into D satisfyingj/(0) = 0, we have KD(z, z) ^ KD g z)y g z)) because KE(ζ, ζ) ^…
Theorem 3.1). Hence, for any holomorphic mapping g(z) of D into D satisfyingj/(0) = 0, we have KD(z, z) ^ KD{g{z)y g{z)) because KE(ζ, ζ) ^ KE(Φ(ζ), Φ(ζ)) under the holomorphic mapping Φ(ζ) = ^ [ ^ ( ζ ) ) ] , Φ(0) = 0, of E into .E. Also we have ϋΓ^O, 0) = minze2) KD(z, z) at only the origin. Moreover, for arbitrary points z {i), z {2) eD, if \,φ~ ι{z {2))\ ^ \φ~ 1{z {l))|, then g{z) = is a holomorphic mapping of Z) into D satisfying g(0) = 0 and 5:
THEOREM 2.1.
THEOREM 2.1. Let D he a hounded schlicht domain of the class ££f. Suppose f:D~>C n is holomorphic, /(0) = 0, and det (df/dz) =N= 0 for all…
THEOREM 2.1. Let D he a hounded schlicht domain of the class ££f. Suppose f:D~>C n is holomorphic, /(0) = 0, and det (df/dz) =N= 0 for all ze D. Then f is a univalent map of D onto a convex domain if
THEOREM 2.2.
THEOREM 2.2. Let D be the unit hypersphere and let f:D-+ C n be holomorphic, /(0) = 0 and det (df/dz) Φ 0 for all zeD. Then f(D) is convex…
THEOREM 2.2. Let D be the unit hypersphere and let f:D-+ C n be holomorphic, /(0) = 0 and det (df/dz) Φ 0 for all zeD. Then f(D) is convex if and only if (2.17) ^rΓ| Az| 2 + z*(&X l^4(Az x Az)Λ ^ 0 , L \dz J dz
THEOREM 2.3.
THEOREM 2.3. Lei D be the polydisk and let f:D->C n be holomorphic, /(0) = 0 and det (df/dz) ^ 0 for all zeD. Then f is a univalent map of…
THEOREM 2.3. Lei D be the polydisk and let f:D->C n be holomorphic, /(0) = 0 and det (df/dz) ^ 0 for all zeD. Then f is a univalent map of D onto a convex domain if and only if the con- dition (2.26) is fulfilled.
THEOREM 3.1.
THEOREM 3.1. Let D he a hounded schlicht domain for which the kernel function KD(z, z) hecomes infinite everywhere on the houndary, KD(0,…
THEOREM 3.1. Let D he a hounded schlicht domain for which the kernel function KD(z, z) hecomes infinite everywhere on the houndary, KD(0, 0) = minZeD KD(z, z) at only the origin, and KD(z, z) ^ KD(g(z), g(z)) for any holomorphic mapping g(z) of D into D satisfying g(0) = 0. Suppose f: D—+C n is holomorphic, /(0) = 0 and det (df/dz) =N= 0 for all zeD. Then f is starlike if and only if for all zeD,z=^ 0.
COROLLARY 3.1.
COROLLARY 3.1. Let D be the unit hypersphere, and let f:D~* C n be holomorphicy /(0) = 0 and det (df/dz) =N= 0 for all zeD. Then f(z) is…
COROLLARY 3.1. Let D be the unit hypersphere, and let f:D~* C n be holomorphicy /(0) = 0 and det (df/dz) =N= 0 for all zeD. Then f(z) is starlike if and only if (3.4) &\z*(MX lf\ > 0 L \ dz / J for all zeD, z^0.
Theorem 4
Theorem 4 [11]:/ = Jw, we^2 are the same as (3.4).
Theorem 4 [11]:/ = Jw, we^2 are the same as (3.4).