Abstract:For holomorphic functions $f$ in the unit disk $ \mathbb{D} $ with $f(0)= 0$, we prove a modulus growth bound involving the logarithmic capacity (transfinite diameter) of the image. We show that the pertinent extremal functions map the unit disk conformally onto the interior of an ellipse. We prove a modulus growth bound for elliptically schlicht functions in terms of the elliptic capacity ${\mathrm{d} }_{\mathrm{e} } f( \mathbb{D} )$ of the image. We also show that the function ${\mathrm{d} }_{\mathrm{e} } f(… Show more
“…The article [6] was a great source of inspiration for several mathematicians to prove monotonicity theorems. Such results can be found in [4], [7], [11], [27] and references therein.…”
Section: Introduction and Statement Of The Resultssupporting
confidence: 58%
“…See for example [3], [4], [5], [6], [7], [8], [9], [11], [16,Chapter 4]. where the at most countably infinite set {z 1 (w), z 2 (w), .…”
Section: Introduction and Statement Of The Resultsmentioning
confidence: 99%
“…Now let w 0 ∈ C \ {0} and let Ω be the doubly connected domain with complementary components [0, w 0 ] and [0, w 0 ] * . The domain Ω can be mapped conformally onto the annulus {z : [4], [12]. The pre-images of the circles |z| = r, where d 0 < r < d Let f : D → C be a nonconstant, elliptically schlicht holomorphic function with f (0) = 0.…”
Section: Introduction and Statement Of The Resultsmentioning
confidence: 99%
“…The pre-images of the circles |z| = r, where d 0 < r < d Let f : D → C be a nonconstant, elliptically schlicht holomorphic function with f (0) = 0. Shah [24] and Betsakos [4] proved respectively that…”
Section: Introduction and Statement Of The Resultsmentioning
confidence: 99%
“…In [4] it was proved that if f : D → C is an elliptically schlicht, holomorphic function, then the function ϕ e (r) = d e (f (rD)) r , 0 < r < 1, is increasing. In view of the elliptic radius and elliptic capacity monotonicity theorems for the elliptically schlicht functions it is natural to wonder whether a similar monotonicity behaviour holds in the context of the elliptic diameter this time (as in [6] and [7]).…”
Abstract. Let f be a holomorphic function of the unit disc D, with f (D) ⊂ D and f (0) = 0. Littlewood's generalization of Schwarz's lemma asserts that for every w ∈ f (D), we have |w| ≤ j |z j |, where {z j } j are the pre-images of w. We consider elliptically schlicht functions and we prove an analogous bound involving the elliptic capacity of the image. For these functions, we also study monotonicity theorems involving the elliptic radius and elliptic diameter.
“…The article [6] was a great source of inspiration for several mathematicians to prove monotonicity theorems. Such results can be found in [4], [7], [11], [27] and references therein.…”
Section: Introduction and Statement Of The Resultssupporting
confidence: 58%
“…See for example [3], [4], [5], [6], [7], [8], [9], [11], [16,Chapter 4]. where the at most countably infinite set {z 1 (w), z 2 (w), .…”
Section: Introduction and Statement Of The Resultsmentioning
confidence: 99%
“…Now let w 0 ∈ C \ {0} and let Ω be the doubly connected domain with complementary components [0, w 0 ] and [0, w 0 ] * . The domain Ω can be mapped conformally onto the annulus {z : [4], [12]. The pre-images of the circles |z| = r, where d 0 < r < d Let f : D → C be a nonconstant, elliptically schlicht holomorphic function with f (0) = 0.…”
Section: Introduction and Statement Of The Resultsmentioning
confidence: 99%
“…The pre-images of the circles |z| = r, where d 0 < r < d Let f : D → C be a nonconstant, elliptically schlicht holomorphic function with f (0) = 0. Shah [24] and Betsakos [4] proved respectively that…”
Section: Introduction and Statement Of The Resultsmentioning
confidence: 99%
“…In [4] it was proved that if f : D → C is an elliptically schlicht, holomorphic function, then the function ϕ e (r) = d e (f (rD)) r , 0 < r < 1, is increasing. In view of the elliptic radius and elliptic capacity monotonicity theorems for the elliptically schlicht functions it is natural to wonder whether a similar monotonicity behaviour holds in the context of the elliptic diameter this time (as in [6] and [7]).…”
Abstract. Let f be a holomorphic function of the unit disc D, with f (D) ⊂ D and f (0) = 0. Littlewood's generalization of Schwarz's lemma asserts that for every w ∈ f (D), we have |w| ≤ j |z j |, where {z j } j are the pre-images of w. We consider elliptically schlicht functions and we prove an analogous bound involving the elliptic capacity of the image. For these functions, we also study monotonicity theorems involving the elliptic radius and elliptic diameter.
Let f : D → C be an analytic function on the unit disc D, with image satisfying a measure condition. We prove an upper and a lower bound for the modulus |w|, for every w ∈ f (D), involving all the pre-images of w.
Στην παρούσα διδακτορική διατριβή μελετήσαμε αρχικά μια γνωστή κλάση μερόμορφων συναρτήσεων, αποδεικνύοντας μονοτονιακά θεωρήματα για συναρτήσεις που περιλαμβάνουν γεωμετρικά και δυναμοθεωρητικά μεγέθη. Στη συνέχεια αποδείξαμε θεωρήματα αυξητικότητας για ολόμορφες συναρτήσεις, με γεωμετρικές συνθήκες για τις εικόνες τους. Επίσης μελετήσαμε την κλάση των ελλειπτικά απλών συναρτήσεων, δίνοντας εκτιμήσεις αυξητικότητας. Τέλος δώσαμε κάποιες γεωμετρικές εκδοχές του γνωστού λήμματος Schwarz.
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