Arunmaran, M.2025-11-172025-11-172023-11-03Proceedings of the Postgraduate Institute of Science Research Congress (RESCON) -2023, University of Peradeniya, P32978-955-8787-09-0https://ir.lib.pdn.ac.lk/handle/20.500.14444/6710The h-function or harmonic-measure distribution function for a region 𝛺 with a specific fixed point zₒ 𝛺, gives some information about the shape of the region. For a region 𝛺 with a basepoint zₒ, we identify the set 𝐸ᵣ, which is the intersection of the boundary of 𝛺 and the closed disc of radius 𝑟 centred at zₒ. The ℎ-function ℎ(𝑟) is given by the harmonic measure of the set 𝐸ᵣ, in 𝛺 at zₒ. This function ℎ(𝑟) only takes the values in the unit interval [0,1], but ℎ(𝑟) will take the value one only for the regions with bounded boundaries. This study is focused on the behaviour of the ℎ-function(s) of a bounded simply connected regions 𝛺 formed by deleting double slits from the unit disc centred at the origin when both slits lie on the real axis and vary in length. Three cases are considered: keep the size of both slits the same; keep the length of one of two slits as the radius of the disc; keep the length of one of two slits as it is bigger than the radius of the disc. For these regions 𝛺, the ℎ-function will take the value 1 after some values of 𝑟. That is, the ℎ-function ℎ(𝑟) meets the line 𝑦= 1 at an angle 𝜓, subtended at the line 𝑦 = 1 with the graph of ℎ(𝑟) in the counterclockwise direction. We check how this angle 𝜓 changes for the above three cases when the basepoint zₒ lies anywhere in between both slits inside the disc. For the first and third cases, when the basepoint zₒ moves between two slits from left to right along the real axis, the angle 𝜓 increases from zero for a while and attains its maximum and then decreases to zero. In the first case, the maximum of the angle 𝜓 has been attained at 𝜋/2. For the second case, when the basepoint zₒ moves between the two slits from left to right along the real line, the angle 𝜓𝜓 decreases from 𝜋/2 to zero. These findings indicate that the ℎ-function of these bounded regions 𝛺 has interesting behaviour at the point 𝑟∗ which is the furthest distance between the base point and the boundary of the region. Future research will focus on checking the behaviour at the same point 𝑟∗, when the basepoint varies along the imaginary axis within these regions 𝛺.en-USBounded regionℎ-functionHarmonic-measureSimply connected regionArticle