L-Index in Joint Variables: Sum and Composition of an Entire Function with a Function With a Vanished Gradient

Bandura, Andriy; Сало, Т. М.; Скасків, О. Б.

doi:10.3390/fractalfract7080593

Cited by 4 publications

(2 citation statements)

References 31 publications

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“…Moreover, the functions of bounded l-index (and bounded index, if l ≡ 1) have applications in the analytic theory of differential equations [15,[28][29][30] and the value distribution theory [17,22,24]. One-dimensional Sheremeta-Kuzyk's approach developed in two multidimensional subapproaches: bounded L-index in direction [9] and bounded L-index in joint variables [10]. A notion of bounded index for bivariate entire functions [23,25]) matches with the notion of bounded L-index in joint variables, if L ≡ (1, .…”

Section: Introductionmentioning

confidence: 99%

Analytic in the unit polydisc functions of bounded L-index in direction

Bandura,

Salo

2023

Mat. Stud.

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The concept of bounded $L$-index in a direction $\mathbf{b}=(b_1,\ldots,b_n)\in\mathbb{C}^n\setminus\{\mathbf{0}\}$ is generalized for a class of analytic functions in the unit polydisc, where $L$ is some continuous function such that for every $z=(z_1,\ldots,z_n)\in\mathbb{D}^n$ one has $L(z)>\beta\max_{1\le j\le n}\frac{|b_j|}{1-|z_j|},$ $\beta=\mathrm{const}>1,$ $\mathbb{D}^n$ is the unit polydisc, i.e. $\mathbb{D}^n=\{z\in\mathbb{C}^n: |z_j|\le 1, j\in\{1,\ldots,n\}\}.$ For functions from this class we obtain sufficient and necessary conditions providing boundedness of $L$-index in the direction. They describe local behavior of maximum modulus of derivatives for the analytic function $F$ on every slice circle $\{z+t\mathbf{b}: |t|=r/L(z)\}$ by their values at the center of the circle, where $t\in\mathbb{C}.$ Other criterion describes similar local behavior of the minimum modulus via the maximum modulus for these functions. We proved an analog of the logarithmic criterion desribing estimate of logarithmic derivative outside some exceptional set by the function $L$. The set is generated by the union of all slice discs $\{z^0+t\mathbf{b}: |t|\le r/L(z^0)\}$, where $z^0$ is a zero point of the function $F$. The analog also indicates the zero distribution of the function $F$ is uniform over all slice discs. In one-dimensional case, the assertion has many applications to analytic theory of differential equations and infinite products, i.e. the Blaschke product, Naftalevich-Tsuji product. Analog of Hayman's Theorem is also deduced for the analytic functions in the unit polydisc. It indicates that in the definition of bounded $L$-index in direction it is possible to remove the factorials in the denominators. This allows to investigate properties of analytic solutions of directional differential equations.

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Section: Introductionmentioning

confidence: 99%

Analytic in the unit polydisc functions of bounded L-index in direction

Bandura,

Salo

2023

Mat. Stud.

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“…Íàñàìêiíåöü çàçíà÷èìî, ùî Ì. Ì. Øåðåìåòà [54,96,106] òàêîae ó ðàìêàõ òåîðiô óíêöié îáìåaeåíîãî iíäåêñó ïåðøèì ïî÷àâ âèâ÷àòè óìîâè, çà ÿêèõ êîìïîçèöiÿ àíà-ëiòè÷íèõ ôóíêöié ìàòèìå îáìåaeåíèé iíäåêñ. Öÿ åñòàôåòà äîñëiäaeåííÿ êîìïîçèöié ó áàãàòîâèìiðíîìó âèïàäêó âiä íüîãî ïiäõîïëåíà ó ñòàòòÿõ [28,30,43].…”

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Theory of Analytic Functions of Bounded Index: M. M. Sheremeta's Ideas and Its Further Development in a Multidimensional Case

Bandura,

Skaskiv

2022

Visnyk of the Lviv Univ. Series Mech. Math.

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On an attempt to introduce a notion of bounded index for the Fueter regular functions of the quaternionic variable

Baksa,

Bandura

2023

Mat. Stud.

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There is introduced a concept of index for the Fueter regular function of the quaternionic variables. There are considered three approaches (Fueter, Sudbery and Mariconda) constructing the Fueter regular function from a holomorphic function of complex variable. Using Mariconda's approach there are constucted some analogs of such elementary functions as the exponent, the sine and the cosine. For the Mariconda analogs we proved that they have bounded index and their indices equal 1, 2, 2, respectively. Using recent results on sum of entire functions whose derivatives are of bounded index it is established that the Fueter regular function constructed by Mariconda's approach is of bounded index, if the derivatives of its addends have bounded index. Also there was examined a function of the form $H(q)=f_1(x_0+ix_1)+jf_2(x_2+ix_3)$, where $f_1$ and $f_2$ are entire functions of complex variable. For the function $H$ it is proved its Fueter regularity and index boundedness if the first order derivatives of $f_1$ and $f_2$ have bounded index. Moreover, the index of the function $H$ does not exceed the maximum of indices of the functions $f'_1$ and $f'_2$ increased by $1$.

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L-Index in Joint Variables: Sum and Composition of an Entire Function with a Function With a Vanished Gradient

Cited by 4 publications

References 31 publications

Analytic in the unit polydisc functions of bounded L-index in direction

Analytic in the unit polydisc functions of bounded L-index in direction

Theory of Analytic Functions of Bounded Index: M. M. Sheremeta's Ideas and Its Further Development in a Multidimensional Case

On an attempt to introduce a notion of bounded index for the Fueter regular functions of the quaternionic variable

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