Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 382 https://internationalpubls.com Subclasses on Negative Coefficints by Linear Differential Operator Annapoorna S.* and Dileep L.* *Vidyavardhaka College Of Engineering, Mysuru, India-570 002 Visvesvaraya Technological University, Belagavi Article History: Received: 12-01-2025 Revised: 15-02-2025 Accepted: 01-03-2025 Abstract: In this paper, we use the linear operator π΄π‘†πœ†,π‘ž 𝛿,𝑛 to define the class 𝑇𝑛(𝛼, 𝛽, 𝛿, πœ†; π‘ž). We derive coefficient estimates and numerous other features for functions that fall under this class. We identify the extreme points and integral means as well. Keywords: Analytic Function, Linear Differential Operator, Coefficient inequalities, Extreme Points and Integral means. AMS Classification: Primary 30C45; Secondary 30C50;30C80 1. Introduction Linear differential operators are crucial in geometric function theory, a branch of mathematics that studies the properties of functions and their translations in geometric contexts. In particular, linear differential operators are used to study the characteristics of conformal mappings, quasi-conformal mappings and other types of mappings between Riemann surfaces and other geometric objects. Numerous aspects of functions and mappings, including their regularity, smoothness, and geometric features like curvature and conformality, are explored using linear operators. Let 𝐴 be the class of functions 𝑓 of the form 𝑓(𝑧) = 𝑧 + βˆ‘ π‘Žπ‘—π‘§π‘—βˆž 𝑗=2 , (1.1) which are analytic in the open unit disc π‘ˆ = {𝑧 ∈ 𝐢; |𝑧| < 1}. Let 𝑇 denote the subclass of 𝐴 in π‘ˆ, consisting of analytic functions whose non-zero coefficients from the second terms onwards are negative. That is, an analytic function 𝑓 ∈ 𝑇 if it has Taylor series expansion of the form 𝑓(𝑧) = 𝑧 βˆ’ βˆ‘ π‘Žπ‘—π‘§π‘—βˆž 𝑗=2 , (π‘Žπ‘— β‰₯ 0) (1.2) which are univalent in the open unit disc π‘ˆ. Using the idea of convolution, Annapoorna S and Dileep L [4], introduced a Linear differential operator 𝑨𝑺𝝀,𝒒 𝜹,𝒏 ∢ 𝑨 β†’ 𝑨 defined by 𝑨𝑺𝝀,𝒒 𝜹,𝒏 𝒇(𝒛) = [(𝟏 βˆ’ 𝝀)[𝟏 + (𝒋 βˆ’ 𝟏)𝜹]𝒏 + π€πš½(𝒂, 𝒄)] βˆ— 𝒇(𝒛) . For Functions 𝑓 ∈ 𝐴 of the form (1.1), we have 𝑨𝑺𝝀,𝒒 𝜹,𝒏 𝒇(𝒛) = 𝑧 + βˆ‘ π΅πœ† 𝛿(π‘Ž, 𝑐, 𝑗, 𝑛; π‘ž)π‘Žπ‘— 𝑧 π‘—βˆž 𝑗=2 (1.3) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 383 https://internationalpubls.com where π΅πœ† 𝛿(π‘Ž, 𝑐, 𝑗, 𝑛; π‘ž) = [(𝟏 βˆ’ 𝝀)[𝟏 + (𝒋 βˆ’ 𝟏)𝜹]𝒏 + 𝝀 (𝒂)π’‹βˆ’πŸ (𝒄)π’‹βˆ’πŸ ] π‘ž (1.4) 𝑛 ∈ β„•0, πœ† β‰₯ 0, 𝛿 β‰₯ 0 π‘Žπ‘›π‘‘ π‘Ž, 𝑐 ∈ ℝ \β„€. Here (π‘Ž)𝑗 is the Pochhammer symbol defined interms of the Gamma function by, (π‘Ž)𝑗 = Ξ“(π‘Ž+𝑗) Ξ“(π‘Ž) = { 1, π‘“π‘œπ‘Ÿ 𝑗 = 0 π‘Ž(π‘Ž + 1)(π‘Ž + 2) β‹― (π‘Ž + 𝑗 βˆ’ 1), π‘“π‘œπ‘Ÿ 𝑗 ∈ β„• . We Obtain the AL-Oboudi differential operator [2], for a range of parametric values of π‘ž β†’ 1βˆ’, πœ† = 0. The Carlson-Shaffer operator [5], is obtained for a range of parametric values of π‘ž β†’ 1βˆ’, πœ† = 1. For a varied parametric values of πœ† = 0, we get the differential operator investigated by Dileep L and Mallige Rajeev [9]. We obtain the operator studied by Dileep L and S Latha [8], for π‘ž β†’ 1βˆ’, 𝛿 = 1. Now using linear differential operator 𝑨𝑺𝝀,𝒒 𝜹,𝒏, we define the following subclass of 𝑇. Let 𝑻𝒏(𝜢, 𝜷, 𝜹, 𝝀; 𝒒) be the subclass of 𝑇 consisting of functions which satisfy the conditions 𝑅 { 𝑧(𝑨𝑺𝝀,𝒒 𝜹,𝒏𝑓) β€² 𝛽𝑧 (𝑨𝑺𝝀,𝒒 𝜹,𝒏𝒇) β€² +(πŸβˆ’πœ·)𝑨𝑺𝝀,𝒒 𝜹,𝒏𝒇 } > 𝛼, (1.5) for some 𝛼, 𝛽 ( 0 ≀ 𝛼, 𝛽 < 1) π‘Žπ‘›π‘‘ 𝑛 ∈ β„•0. For a different parametric values of π‘ž β†’ 1βˆ’, 𝛿 = 1 π‘Žπ‘›π‘‘ πœ† = 0 the above class reduces to the class defined by Dileep L and S Latha [8]. 2. Prime Results: Theorem 2.1: A function 𝑓 defined by (1.2) is in the class 𝑻𝒏(𝜢, 𝜷, 𝜹, 𝝀; 𝒒) if and only if βˆ‘ π΅πœ† 𝛿(π‘Ž, 𝑐, 𝑗, 𝑛; π‘ž)π‘Žπ‘— [𝑗 βˆ’ 𝛼 + 𝛼𝛽 βˆ’ 𝛼𝛽𝑗] < 1 βˆ’ 𝛼,∞ 𝑗=2 (2.1) where, 𝛼, 𝛽 ( 0 ≀ 𝛼, 𝛽 < 1) π‘Žπ‘›π‘‘ 𝑛 ∈ β„•0. Proof: Suppose 𝑓 ∈ 𝑻𝒏(𝜢, 𝜷, 𝜹, 𝝀; 𝒒). Then 𝑅 { 𝑧(𝑨𝑺𝝀,𝒒 𝜹,𝒏𝑓)β€² 𝛽𝑧 (𝑨𝑺𝝀,𝒒 𝜹,𝒏𝒇) β€² + (𝟏 βˆ’ 𝜷)𝑨𝑺𝝀,𝒒 𝜹,𝒏𝒇 } > 𝛼 𝑅 { 𝑧 βˆ’ βˆ‘ π‘—π΅πœ† 𝛿(π‘Ž, 𝑐, 𝑗, 𝑛; π‘ž)π‘Žπ‘— 𝑧 π‘—βˆž 𝑗=2 𝛽 [𝑧 βˆ’ βˆ‘ π‘—π΅πœ† 𝛿(π‘Ž, 𝑐, 𝑗, 𝑛; π‘ž)π‘Žπ‘— π‘§π‘—βˆž 𝑗=2 ] + (𝟏 βˆ’ 𝜷)[𝑧 βˆ’ βˆ‘ π΅πœ† 𝛿(π‘Ž, 𝑐, 𝑗, 𝑛; π‘ž)π‘Žπ‘— π‘§π‘—βˆž 𝑗=2 ] } > 𝛼 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 384 https://internationalpubls.com 𝑅 { π‘§βˆ’βˆ‘ π‘—π΅πœ† 𝛿(π‘Ž,𝑐,𝑗,𝑛;π‘ž)π‘Žπ‘— 𝑧 π‘—βˆž 𝑗=2 π’›βˆ’βˆ‘ π΅πœ† 𝛿(π‘Ž,𝑐,𝑗,𝑛;π‘ž)[𝛽(π‘—βˆ’1)+1]π‘Žπ‘— 𝑧 π‘—βˆž 𝑗=2 } > 𝛼. Let 𝑧 β†’ 1, then we get 1 βˆ’ βˆ‘ π‘—π΅πœ† 𝛿(π‘Ž, 𝑐, 𝑗, 𝑛; π‘ž)π‘Žπ‘— ∞ 𝑗=2 > 𝛼 {1 βˆ’ βˆ‘ π΅πœ† 𝛿(π‘Ž, 𝑐, 𝑗, 𝑛; π‘ž)[𝛽(𝑗 βˆ’ 1) + 1]π‘Žπ‘— ∞ 𝑗=2 } βˆ‘ π‘—π΅πœ† 𝛿(π‘Ž, 𝑐, 𝑗, 𝑛; π‘ž)π‘Žπ‘— ∞ 𝑗=2 βˆ’ 𝛼 βˆ‘ π΅πœ† 𝛿(π‘Ž, 𝑐, 𝑗, 𝑛; π‘ž)[𝛽(𝑗 βˆ’ 1) + 1]π‘Žπ‘— ∞ 𝑗=2 < 1 βˆ’ 𝛼 βˆ‘ π΅πœ† 𝛿(π‘Ž, 𝑐, 𝑗, 𝑛; π‘ž)π‘Žπ‘— [𝑗 βˆ’ 𝛼 + 𝛼𝛽 βˆ’ 𝛼𝛽𝑗] < 1 βˆ’ 𝛼. ∞ 𝑗=2 Conversely, assume that (2.1) be true. We have to show that (1.5) is satisfied or equivalently | 𝑧(𝑨𝑺𝝀,𝒒 𝜹,𝒏𝑓)β€² 𝛽𝑧 (𝑨𝑺𝝀,𝒒 𝜹,𝒏𝒇) β€² + (𝟏 βˆ’ 𝜷)𝑨𝑺𝝀,𝒒 𝜹,𝒏𝒇 βˆ’ 1| < 1 βˆ’ 𝛼. But | 𝑧 βˆ’ βˆ‘ π‘—π΅πœ† 𝛿(π‘Ž, 𝑐, 𝑗, 𝑛; π‘ž)π‘Žπ‘— 𝑧 π‘—βˆž 𝑗=2 𝒛 βˆ’ βˆ‘ π΅πœ† 𝛿(π‘Ž, 𝑐, 𝑗, 𝑛; π‘ž)[𝛽(𝑗 βˆ’ 1) + 1]π‘Žπ‘— π‘§π‘—βˆž 𝑗=2 βˆ’ 1| = | βˆ‘ π΅πœ† 𝛿(π‘Ž, 𝑐, 𝑗, 𝑛; π‘ž)π‘Žπ‘— (𝑗 βˆ’ 1)(𝛽 βˆ’ 1)π‘§π‘—βˆž 𝑗=2 𝒛 βˆ’ βˆ‘ π΅πœ† 𝛿(π‘Ž, 𝑐, 𝑗, 𝑛; π‘ž)[𝛽(𝑗 βˆ’ 1) + 1]π‘Žπ‘— π‘§π‘—βˆž 𝑗=2 | ≀ βˆ‘ π΅πœ† 𝛿(π‘Ž,𝑐,𝑗,𝑛;π‘ž)π‘Žπ‘— (π‘—βˆ’1)(π›½βˆ’1)|𝑧𝑗|∞ 𝑗=2 |𝒛|βˆ’βˆ‘ π΅πœ† 𝛿(π‘Ž,𝑐,𝑗,𝑛;π‘ž)[𝛽(π‘—βˆ’1)+1]π‘Žπ‘— |𝑧 𝑗|∞ 𝑗=2 ≀ βˆ‘ π΅πœ† 𝛿(π‘Ž,𝑐,𝑗,𝑛;π‘ž)π‘Žπ‘— (π‘—βˆ’1)(π›½βˆ’1)∞ 𝑗=2 1βˆ’βˆ‘ π΅πœ† 𝛿(π‘Ž,𝑐,𝑗,𝑛;π‘ž)[𝛽(π‘—βˆ’1)+1]π‘Žπ‘— ∞ 𝑗=2 . The last expression is bounded by 1 βˆ’ 𝛼 if βˆ‘ π΅πœ† 𝛿(π‘Ž, 𝑐, 𝑗, 𝑛; π‘ž)π‘Žπ‘— (𝑗 βˆ’ 1)(𝛽 βˆ’ 1) ∞ 𝑗=2 ≀ (1 βˆ’ 𝛼) (1 βˆ’ βˆ‘ π΅πœ† 𝛿(π‘Ž, 𝑐, 𝑗, 𝑛; π‘ž)[𝛽(𝑗 βˆ’ 1) + 1]π‘Žπ‘— ∞ 𝑗=2 ) βˆ‘ π΅πœ† 𝛿(π‘Ž, 𝑐, 𝑗, 𝑛; π‘ž)π‘Žπ‘— [𝑗 βˆ’ 𝛼 + 𝛼𝛽 βˆ’ 𝛼𝛽𝑗] < 1 βˆ’ 𝛼, ∞ 𝑗=2 which is true by hypothesis. This completes the assertion of Theorem 2.1 For parametric values of π‘ž β†’ 1βˆ’, 𝛿 = 1, πœ† = 0 and different values of 𝑛 (𝑛 = 0, 1) in the above theorem, we have the following results of A O Mostafa [15]. Corollary 2.2: (π’Š) A function 𝑓 defined by (1.2) is in the class 𝑻(𝜢, 𝜷) if and only if Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 385 https://internationalpubls.com βˆ‘ [𝑗 βˆ’ 𝛼 + 𝛼𝛽 βˆ’ 𝛼𝛽𝑗]π‘Žπ‘— ≀ 1 βˆ’ 𝛼. ∞ 𝑗=2 (𝑖𝑖) A function 𝑓 defined by (1.2) is in the class π‘ͺ(𝜢, 𝜷) if and only if βˆ‘ 𝑗[𝑗 βˆ’ 𝛼 + 𝛼𝛽 βˆ’ 𝛼𝛽𝑗]π‘Žπ‘— ≀ 1 βˆ’ 𝛼. ∞ 𝑗=2 Corollary 2.3: If 𝑓 ∈ 𝑻𝒏(𝜢, 𝜷, 𝜹, 𝝀; 𝒒), then |π‘Žπ‘—| ≀ 1βˆ’π›Ό π΅πœ† 𝛿(π‘Ž,𝑐,𝑗,𝑛;π‘ž)[π‘—βˆ’π›Ό+π›Όπ›½βˆ’π›Όπ›½π‘—] . Theorem 2.4: Let 0 ≀ 𝛼 < 1, 0 ≀ 𝛽1 ≀ 𝛽2 < 1 , 𝑛 ∈ 𝑁0, then 𝑻𝒏(𝜢, 𝜷𝟐, 𝜹, 𝝀; 𝒒) βŠ‚ 𝑻𝒏(𝜢, 𝜷𝟏, 𝜹, 𝝀; 𝒒). Proof: From the Theorem 2.1, βˆ‘ π΅πœ† 𝛿(π‘Ž, 𝑐, 𝑗, 𝑛; π‘ž)π‘Žπ‘— [𝑗 βˆ’ 𝛼 + 𝛼𝛽2 βˆ’ 𝛼𝛽2𝑗] ∞ 𝑗=2 ≀ βˆ‘ π΅πœ† 𝛿(π‘Ž, 𝑐, 𝑗, 𝑛; π‘ž)π‘Žπ‘— [𝑗 βˆ’ 𝛼 + 𝛼𝛽1 βˆ’ 𝛼𝛽1𝑗]∞ 𝑗=2 ≀ 1 βˆ’ 𝛼. For 𝑓(𝑧) ∈ 𝑻𝒏(𝜢, 𝜷𝟐, 𝜹, 𝝀; 𝒒). Hence 𝑓(𝑧) ∈ 𝑻𝒏(𝜢, 𝜷𝟏, 𝜹, 𝝀; 𝒒). Theorem 2.5: Let 𝑓(𝑧) ∈ 𝑻𝒏(𝜢, 𝜷, 𝜹, 𝝀; 𝒒). Define 𝑓1(𝑧) = 𝑧 and 𝒇𝒋(𝒛) = 𝒛 + 1 βˆ’ 𝛼 π΅πœ† 𝛿(π‘Ž, 𝑐, 𝑗, 𝑛; π‘ž)[𝑗 βˆ’ 𝛼 + 𝛼𝛽 βˆ’ 𝛼𝛽𝑗] 𝑧𝑗 , 𝑗 = 2, 3, β‹―, for some 𝛼, 𝛽 (0 ≀ 𝛼, 𝛽 < 1 ), 𝑛 ∈ β„•0 π‘Žπ‘›π‘‘ 𝑧 ∈ π‘ˆ. 𝑓 ∈ 𝑻𝒏(𝜢, 𝜷, 𝜹, 𝝀; 𝒒) if and only if 𝑓 can be expressed as 𝑓(𝑧) = βˆ‘ πœ‡π‘— ∞ 𝑗=1 𝑓𝑗(𝑧) where πœ‡π‘— β‰₯ 0 and βˆ‘ πœ‡π‘— ∞ 𝑗=1 = 1. Proof: If 𝑓(𝑧) = βˆ‘ πœ‡π‘— ∞ 𝑗=1 𝑓𝑗(𝑧) with βˆ‘ πœ‡π‘— ∞ 𝑗=1 = 1, πœ‡π‘— β‰₯ 0, then βˆ‘ π΅πœ† 𝛿(π‘Ž, 𝑐, 𝑗, 𝑛; π‘ž)[𝑗 βˆ’ 𝛼 + 𝛼𝛽 βˆ’ 𝛼𝛽𝑗] πœ‡π‘— π΅πœ† 𝛿(π‘Ž, 𝑐, 𝑗, 𝑛; π‘ž)[𝑗 βˆ’ 𝛼 + 𝛼𝛽 βˆ’ 𝛼𝛽𝑗] ∞ 𝑗=2 (1 βˆ’ 𝛼) = βˆ‘ πœ‡π‘— (1 βˆ’ 𝛼) = (1 βˆ’ πœ‡1)(1 βˆ’ 𝛼) ≀ (1 βˆ’ 𝛼). ∞ 𝑗=2 Hence 𝑓(𝑧) ∈ 𝑻𝒏(𝜢, 𝜷, 𝜹, 𝝀; 𝒒). Conversely, let 𝑓(𝑧) = 𝑧 βˆ’ βˆ‘ π‘Žπ‘—π‘§π‘—βˆž 𝑗=2 ∈ 𝑻𝒏(𝜢, 𝜷, 𝜹, 𝝀; 𝒒), define 𝝁𝒋 = π΅πœ† 𝛿(π‘Ž,𝑐,𝑗,𝑛;π‘ž)[π‘—βˆ’π›Ό+π›Όπ›½βˆ’π›Όπ›½π‘—]|π‘Žπ‘—|, (πŸβˆ’πœΆ) 𝒋 = 𝟐, πŸ‘ β‹―, and define πœ‡1 = 1 βˆ’ βˆ‘ πœ‡π‘—.∞ 𝑗=2 From Theorem 2.1, βˆ‘ πœ‡π‘— ≀ 1∞ 𝑗=2 π‘Žπ‘›π‘‘ π‘ π‘œ πœ‡1 β‰₯ 0. Since πœ‡π‘—π‘“π‘—(𝑧) = πœ‡π‘—π‘“ + π‘Žπ‘— 𝑧𝑗 , βˆ‘ πœ‡π‘— 𝑓𝑗(𝑧) = 𝑧 βˆ’ βˆ‘ π‘Žπ‘—π‘§π‘—βˆž 𝑗=2 = 𝑓(𝑧).∞ 𝑗=1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 386 https://internationalpubls.com Theorem 2.6: The class 𝑻𝒏(𝜢, 𝜷, 𝜹, 𝝀; 𝒒) is closed under convex linear combination. Proof: Let 𝑓, 𝑔 ∈ 𝑻𝒏(𝜢, 𝜷, 𝜹, 𝝀; 𝒒) and let 𝑓(𝑧) = 𝑧 βˆ’ βˆ‘ π‘Žπ‘—π‘§π‘— ∞ 𝑗=2 , 𝑔(𝑧) = 𝑧 βˆ’ βˆ‘ 𝑏𝑗𝑧𝑗 ∞ 𝑗=2 . For πœ‚ such that 0 ≀ πœ‚ ≀ 1, it suffices to show that the function defined by β„Ž(𝑧) = (1 βˆ’ πœ‚)𝑓(𝑧) + πœ‚π‘”(𝑧), 𝑧 ∈ π‘ˆ belongs to 𝑻𝒏(𝜢, 𝜷, 𝜹, 𝝀; 𝒒) . Now β„Ž(𝑧) = 𝑧 βˆ’ βˆ‘[(1 βˆ’ πœ‚)π‘Žπ‘— + πœ‚π‘π‘—]𝑧𝑗 , ∞ 𝑗=2 Applying Theorem 2.1, to 𝑓, 𝑔 ∈ 𝑻𝒏(𝜢, 𝜷, 𝜹, 𝝀; 𝒒) we have βˆ‘ π΅πœ† 𝛿(π‘Ž, 𝑐, 𝑗, 𝑛; π‘ž)[𝑗 βˆ’ 𝛼 + 𝛼𝛽 βˆ’ 𝛼𝛽𝑗][(1 βˆ’ πœ‚)π‘Žπ‘— + πœ‚π‘π‘—] ∞ 𝑗=2 = (1 βˆ’ πœ‚) βˆ‘ π΅πœ† 𝛿(π‘Ž, 𝑐, 𝑗, 𝑛; π‘ž)[𝑗 βˆ’ 𝛼 + 𝛼𝛽 βˆ’ 𝛼𝛽𝑗] π‘Žπ‘— + πœ‚ βˆ‘ π΅πœ† 𝛿(π‘Ž, 𝑐, 𝑗, 𝑛; π‘ž)[𝑗 βˆ’ 𝛼 + 𝛼𝛽 βˆ’ 𝛼𝛽𝑗] 𝑏𝑗 ∞ 𝑗=2 ∞ 𝑗=2 ≀ (1 βˆ’ πœ‚)(1 βˆ’ 𝛼) + πœ‚ (1 βˆ’ 𝛼) = 1 βˆ’ 𝛼. This implies that β„Ž ∈ 𝑻𝒏(𝜢, 𝜷, 𝜹, 𝝀; 𝒒). Corollary 2.7: If 𝑓1(𝑧), 𝑓2(𝑧) ∈ 𝑻𝒏(𝜢, 𝜷, 𝜹, 𝝀; 𝒒) then the function defined by π’ˆ(𝒛) = 𝟏 𝟐 [π’‡πŸ(𝒛) + π’‡πŸ(𝒛)] is also in 𝑻𝒏(𝜢, 𝜷, 𝜹, 𝝀; 𝒒). Theorem 2.8: Let for π‘š = 1,2, β‹― , 𝑗 π‘“π‘š(𝑧) = 𝑧 βˆ’ βˆ‘ π‘Žπ‘—,π‘š π‘§π‘—βˆž 𝑗=2 ∈ 𝑻𝒏(𝜢, 𝜷, 𝜹, 𝝀; 𝒒) and 0 < π›½π‘š < 1 such that βˆ‘ π›½π‘š = 1,∞ π‘š=2 then the function 𝐹(𝑧) defined by 𝐹(𝑧) = βˆ‘ π›½π‘šπ‘“π‘š(𝑧)𝑗 π‘š=2 𝑖𝑠 π‘Žπ‘™π‘ π‘œ 𝑖𝑛 𝑻𝒏(𝜢, 𝜷, 𝜹, 𝝀; 𝒒). 𝑷𝒓𝒐𝒐𝒇: For each π‘š ∈ { 1,2, β‹― , 𝑗} we obtain βˆ‘ π΅πœ† 𝛿(π‘Ž, 𝑐, 𝑗, 𝑛; π‘ž)[𝑗 βˆ’ 𝛼 + 𝛼𝛽 βˆ’ 𝛼𝛽𝑗] |π‘Žπ‘—| < 1 βˆ’ 𝛼. ∞ 𝑗=2 𝐹(𝑧) = βˆ‘ π›½π‘š (𝑧 βˆ’ βˆ‘ π‘Žπ‘—,π‘š π‘§π‘—βˆž 𝑗=2 )𝑗 π‘š=1 = 𝑧 βˆ’ βˆ‘ ( βˆ‘ π›½π‘š π‘Žπ‘—,π‘š 𝑧𝑗 𝑗 π‘š=1 ) ∞ 𝑗=2 Since, βˆ‘ π΅πœ† 𝛿(π‘Ž, 𝑐, 𝑗, 𝑛; π‘ž)[𝑗 βˆ’ 𝛼 + 𝛼𝛽 βˆ’ 𝛼𝛽𝑗] [ βˆ‘ π›½π‘š π‘Žπ‘—,π‘š 𝑗 π‘š=1 ] ∞ 𝑗=2 < βˆ‘ 𝛽𝑗(1 βˆ’ 𝛼) < 1 βˆ’ 𝛼 𝑗 π‘š=1 . Therefore, 𝐹(𝑧) ∈ 𝑻𝒏(𝜢, 𝜷, 𝜹, 𝝀; 𝒒). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 387 https://internationalpubls.com Theorem 2.9: Let 𝑓(𝑧) ∈ 𝑻𝒏(𝜢, 𝜷, 𝜹, 𝝀; 𝒒). Komato operator of 𝑓 is defined by π‘˜(𝑧) = ∫ (𝑐 + 1)𝛾 Ξ“(𝛾) 1 0 𝑑𝑐 (log ( 1 𝑑 )) π›Ύβˆ’1 𝑓(𝑑𝑧) 𝑑 𝑑𝑑, 𝑐 > βˆ’1, 𝛾 β‰₯ 0 then π‘˜(𝑧) ∈ 𝑻𝒏(𝜢, 𝜷, 𝜹, 𝝀; 𝒒). 𝑷𝒓𝒐𝒐𝒇: We have ∫ 𝑑𝑐 (log ( 1 𝑑 )) π›Ύβˆ’1 𝑑𝑑 𝟏 𝟎 = πšͺ(𝜸) (𝒄 + 𝟏)𝜸 ∫ 𝑑𝑗+π‘βˆ’1 (log ( 1 𝑑 )) π›Ύβˆ’1 𝑑𝑑 𝟏 𝟎 = πšͺ(𝜸) (𝒄 + 𝟏)𝜸 , 𝒋 = 𝟐, πŸ‘, β‹―, π‘˜(𝑧) = (𝑐 + 1)𝛾 Ξ“(𝛾) [∫ 𝑑𝑐 (log ( 1 𝑑 )) π›Ύβˆ’1 𝑧 𝑑𝑑 βˆ’ βˆ‘ 𝑧𝑗 ∫ π‘Žπ‘— 𝑑𝑗+π‘βˆ’1 (log ( 1 𝑑 )) π›Ύβˆ’1 𝑑𝑑 1 0 ∞ 𝑗=2 1 0 ] = 𝒛 βˆ’ βˆ‘ ( 𝒄 + 𝟏 𝒄 + 𝒋 ) 𝜸 𝒂𝒋𝒛𝒋. ∞ 𝒋=𝟐 Since 𝑓(𝑧) ∈ 𝑻𝒏(𝜢, 𝜷, 𝜹, 𝝀; 𝒒) and ( 𝒄+𝟏 𝒄+𝒋 ) 𝜸 < 𝟏, we have βˆ‘ π΅πœ† 𝛿(π‘Ž, 𝑐, 𝑗, 𝑛; π‘ž)[𝑗 βˆ’ 𝛼 + 𝛼𝛽 βˆ’ 𝛼𝛽𝑗] ( 𝒄 + 𝟏 𝒄 + 𝒋 ) 𝜸 π‘Žπ‘— < (1 βˆ’ 𝛼). ∞ 𝑗=2 Theorem 2.10: Let 𝑓(𝑧) ∈ 𝑻𝒏(𝜢, 𝜷, 𝜹, 𝝀; 𝒒), then for every 0 ≀ 𝜁 < 1 the function π‘―πœ»(𝒛) = (𝟏 βˆ’ 𝜻)𝒇(𝒛) + 𝜻 ∫ 𝒇(𝒕) 𝒕 𝒛 𝟎 𝒅𝒕. 𝑷𝒓𝒐𝒐𝒇: We have 𝐻𝜁(𝑧) = 𝑧 βˆ’ βˆ‘ (1 + 𝜁 𝑗 βˆ’ 𝜁) π‘Žπ‘—π‘§π‘— .∞ 𝑗=2 Since (1 + 𝜁 𝑗 βˆ’ 𝜁) < 1, 𝑗 β‰₯ 2, π‘ π‘œ 𝑏𝑦 Theorem 2.1, βˆ‘ (1 + 𝜁 𝑗 βˆ’ 𝜁) π΅πœ† 𝛿(π‘Ž, 𝑐, 𝑗, 𝑛; π‘ž)[𝑗 βˆ’ 𝛼 + 𝛼𝛽 βˆ’ 𝛼𝛽𝑗]π‘Žπ‘— ∞ 𝑗=2 < βˆ‘ π΅πœ† 𝛿(π‘Ž, 𝑐, 𝑗, 𝑛; π‘ž)[𝑗 βˆ’ 𝛼 + 𝛼𝛽 βˆ’ 𝛼𝛽𝑗]π‘Žπ‘— < 1 βˆ’ 𝛼. ∞ 𝑗=2 Therefore, 𝐻𝜁(𝑧) ∈ 𝑻𝒏(𝜢, 𝜷, 𝜹, 𝝀; 𝒒). 3. 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