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Yayın Asymptotic behavior of the solutions of a transmission problem for the Helmholtz equation: A functional analytic approach(Wiley, 2022) Akyel, Tuğba; Lanza de Cristoforis, MassimoLet omega(i), omega(o) be bounded open connected subsets of Double-struck capital Rn that contain the origin. Let omega(epsilon)equivalent to omega o\epsilon omega i? for small epsilon > 0. Then, we consider a linear transmission problem for the Helmholtz equation in the pair of domains epsilon omega(i) and omega(epsilon) with Neumann boundary conditions on partial differential omega(o). Under appropriate conditions on the wave numbers in epsilon omega(i) and omega(epsilon) and on the parameters involved in the transmission conditions on epsilon partial differential omega(i), the transmission problem has a unique solution (u(i)(epsilon, center dot), u(o)(epsilon, center dot)) for small values of epsilon > 0. Here, u(i)(epsilon, center dot) and u(o)(epsilon, center dot) solve the Helmholtz equation in epsilon omega(i) and omega(epsilon), respectively. Then, we prove that if x is an element of omega(o) \ {0}, then u(o)(epsilon, x) can be expanded into a convergent power expansion of epsilon, kappa n epsilon log epsilon,delta 2,nlog-1 epsilon for epsilon small enough. Here, kappa n=1 if n is even and kappa n=0 if n is odd, and delta(2, 2) equivalent to 1 and delta(2, n) equivalent to 0 if n >= 3.Yayın Estimates for ?-spirallike functions of complex order on the boundary(Ukrainian Mathematical Journal, 2022) Akyel, TuğbaWe give some results obtained for ?-spirallike functions of complex order on the boundary of the unit disc U. The sharpness of these results is also proved. Furthermore, three examples of our results are considered.Yayın Microscopic behavior of the solutions of a transmission problem for the Helmholtz equation: a functional analytic approach(Stud. Univ. Babe¸s-Bolyai Math., 2022) Akyel, Tuğba; Lanza de Cristoforis, MassimoLet Omega(i), Omega(o) be bounded open connected subsets of R-n that contain the origin. Let Omega(epsilon) equivalent to Omega(o) \ c (Omega) over bar (i) for small epsilon > 0. Then we consider a linear transmission problem for the Helmholtz equation in the pair of domains epsilon Omega(i) and Omega(epsilon) with Neumann boundary conditions on partial derivative Omega(o). Under appropriate conditions on the wave numbers in epsilon Omega(i) and Omega(epsilon) and on the parameters involved in the transmission conditions on epsilon partial derivative Omega(i), the transmission problem has a unique solution (u(i)(epsilon, .); u(o) (epsilon, .)) for small values of epsilon > 0. Here u(i)(epsilon, .) and u(o) (epsilon, .) solve the Helmholtz equation in epsilon Omega(i) and Omega(epsilon), respectively. Then we prove that if xi is an element of(Omega(i)) over bar and xi is an element of R-n\Omega(i) then the rescaled solutions u(i) (epsilon, epsilon xi) and u(o) (epsilon, epsilon xi) can be expanded into a convergent power expansion of epsilon, kappa(n) is an element of log epsilon, delta(2,n) log(-1) epsilon, kappa(n) is an element of log(2) epsilon for epsilon small enough. Here kappa(n) = 1 if n is even and kappa(n) = 0 if n is odd and delta(2,2) equivalent to 1 and delta(2,n) equivalent to 0 if n >= 3.Yayın ON BOUNDS FOR THE DERIVATIVE OF ANALYTIC FUNCTIONS AT THE BOUNDARY(Kangwon-Kyungki Mathematical Soc, 2021) Örnek, Bülent Nafi; Akyel, TuğbaIn this paper, we obtain a new boundary version of the Schwarz lemma for analytic function. We give sharp upper bounds for vertical bar f'(0)vertical bar and sharp lower bounds for vertical bar f'(c)vertical bar with c is an element of partial derivative D -{z:vertical bar z vertical bar - 1}. Thus we present some new inequalities for analytic functions. Also, we estimate the modulus of the angular derivative of the function f(z) from below according to the second Taylor coefficients of f about z = 0 and z = z(0) not equal 0. Thanks to these inequalities, we see the relation between vertical bar f'(0)vertical bar and Rf(0). Similarly, we see the relation between Rf(0) and vertical bar f'(c)vertical bar for some c is an element of partial derivative D. The sharpness of these inequalities is also proved.Yayın On the Rigidity Part of Schwarz Lemma(Amer Inst Physics, 2019) Akyel, Tuğba; Örnek, Bülent NafiWe consider the rigidity part of Schwarz Lemma. Let f be a holomorphic function in the unit disc D and vertical bar Rf(z)vertical bar < 1 for vertical bar z vertical bar < 1. We generalize the rigidity of holomorphic function and provide sufficient conditions on the local behaviour of f near a finite set of boundary points that needs f to be a finite Blaschke product. For a different version of the rigidity theorems of D. Burns-S.Krantz and D. Chelst, we present some more general results in which the bilogaritmic concave majorants are used. The strategy of these results relies on a special version of Phragmen-Lindelof princible and Harnack inequality.Yayın On the rigidity part of Schwarz Lemma at the boundary(Maltepe Üniversitesi, 2019) Akyel, Tuğba; Örnek, Bülent NafiWe consider the rigidity part of Schwarz Lemma. Let f be a holomorphic function in the unit disc D and |?f(z)| < 1 for |z| < 1. We generalize rigidity of holomorphic function and provide sufficient conditions on the local behaviour of f near a finite set of boundary points that needs f to be a finite Blaschke product. For a different version of the rigidity theorems of D. Burns-S.Krantz and D. Chelst, we present some more general results used the bilogaritmic concave majorants. The strategy of these results relies on a special version of Phragmen-Lindel¨of princible and Harnack inequalityYayın Some remarks for a certain class of holomorphic functions at the boundary of the unit disc(2019) Akyel, Tuğba; Örnek, Bülent NafiWe consider a boundary version of the Schwarz Lemma on a certain class which is denoted by ??(??). Forthe function ??(??) = ?? + ???????? + ????????+. .. which is defined in the unit disc ?? such that the function ??(??)belongs to the class ??(??), we estimate from below the modulus of the angular derivative of the function????'(??)??(??)at the boundary point ?? with ????'(??)??(??)=????????. Moreover, we get the Schwarz Lemma for the class ??(??).We also investigate some inequalities obtained in terms of sharpness.Yayın SOME REMARKS ON THE SUBORDINATION PRINCIPLE FOR ANALYTIC FUNCTIONS CONCERNED WITH ROGOSINSKI'S LEMMA(Kangwon-Kyungki Mathematical Soc, 2021) Akyel, TuğbaIn this paper, we present a Schwarz lemma at the boundary for analytic functions at the unit disc, which generalizes classical Schwarz lemma for bounded analytic functions. For new inequalities, the results of Rogosinski's lemma, Subordination principle and Jack's lemma were used.Yayın Some results for a certain class of holomorphic functions at the boundary of the unit disc(Maltepe Üniversitesi, 2019) Örnek, Bülent Nafi; Akyel, TuğbaWe consider a version of the boundary Schwarz Lemma on a certain class which is denoted by K(?). For the function f(z) = z + c2z2 + c3z3 + ... defined in the unit disc E such that the function f(z) belongs to the class K(?), we estimate from below the modulus of the angular derivative of the function z f (z) f(z) at the boundary point b with b f (b) f(b) = 1 1+? . Moreover, we get Schwarz Lemma for the class K(?). We also investigate some inequalities obtained in terms of sharpness.Yayın Upper Bound of Hankel Determinant for a Class of Analytic Functions(Univ Nis, Fac Sci Math, 2021) Akyel, TuğbaThe aim of this study is to solve the Fekete-Szego problem and to define upper bound for Hankel determinant H-2(1) in a novel class K of analytical functions in the unit disc. Moreover, in a class of analytic functions on the unit disc, assuming the existence of an angular limit on the boundary point, the estimations from below of the modulus of angular derivative have been obtained.
