Title Science and Technology Indonesia e-ISSN:2580-4391 p-ISSN:2580-4405 Vol. 7, No. 1, January 2022 Research Paper Coefficient Bound of a New Generalized Class of Tilted Analytic Univalent Functions Abdullah Yahya1*, Mohd Najir Tokachil1, Hasilatul Hana Hamzah2, Shahriel Mohd Saifullah Pirman3, Siti Asiah Che Muhammad4 1Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA, Seremban, 70300, Malaysia 2J.K Wire Harness Sdn. Bhd. No. 7 and 9, Kawasan Perindustrian Tebrau 1, Johor Bahru, 81100, Malaysia 3Sekolah Menengah Kebangsaan Jambatan Putih, Tawau, 91019, Malaysia 4Power Cables Malaysia, Purchasing Department, Shah Alam, 40702, Malaysia *Corresponding author: abdullahyahya@uitm.edu.my Abstract This paper is concerned with the new generalized class of tilted analytic univalent functions, S∗(θ,α,s,t) which denoted as. Re{eiθ f ′(z) z m(z) } > α, for cos θ > α, |θ| < π, 0 ≤ α <1, m(z)= z (1−sz)(1−tz) , s, 1, -1 ≤ t < 1, s, t and s, t ∈ C which is analytic in the unit disk ∆={w:|w|<1}. The coefficient bound as well as representation theorem of extremal properties is obtained S∗(θ,α,s,t). Keywords Analytic Functions, Univalent Functions, Representation Theorem, Coefficient Bound Received: 15 September 2021, Accepted: 02 January 2022 https://doi.org/10.26554/sti.2022.7.1.67-72 1. INTRODUCTION Let U represent the class of functions presented by f (z)= z+ a2z 2 + a3z 3 + ...+ anz n + ... (1) which are regular in ∆={w:|w|<1} and is the subclass of U consisting of normalized univalent functions which are satis- fying f(0)=0 and f’(0)-1=0 (Goodman, 1983; Duren, 2001). We also denote the subclass of U containing functions that are starlike, convex, and close-to-convex by St, K and C respec- tively. Silverman and Silvia (1996) obtained representation theorem and coe�cient bound for the class G(δ, 0) satis�es Re(eiδ f ′z h′z ) >0 |δ|< π 2 , h(z) ∈ K and z ∈ ∆. Subsequently, Mo- hamad (1998) introduced the class G (λ,γ) satis�es Re(eiλ f ′z h′z ) > γ,where |λ|≤ π, cosλ- γ >0 and z ∈∆ which is the same h,(z) done by Silverman and Silvia (1996) and discovered represen- tation theorem and coe�cient bound of G (α,δ). Afterward, Soh and Mohamad (2006) determined extremal properties for the class of functions which h(z)= -z-2 (log(1-z)) implies n’(z)= (1+(z))(1-z)−1 where it is the extreme function produced by MacGregor (1962) . Next, Cik Soh (2009) introduced the class G (λ,γ,γ) with the form of kγ(z)= -z(1-2γ)-(log(1-z))(2-2γ) where 0≤γ≤1 g′(z)= 1+(1−2γ)z1−z and the author presented some extremal properties of G (λ,γ,γ). Kaharudin (2011) has ex- panded the results from Cik Soh (2009) by presenting a special case of gβ (z) when β= 1 2 given by Brickman et al. (1973) and found important properties of the coe�cient bound and the ro- tation theorem for this class of functions. In addition, many re- searchers such as MacGregor (1962) , Goel (1967) , MacGregor (1964) , Silverman (1972) , Goel and Mehrok (1983) , Silverman and Telage (1979) , Fukui et al. (1987) , Yahya et al. (2014) , Ka- plan (1952) , Mohamad (2000) , Akbarally et al. (2011) , Wang (2010) , Shashkin (1994) , Çağlar et al. (2013) , Magesh and Yamini (2013) , Elhaddad and Darus (2020) , Adegani et al. (2021) , and Shaba and Wanas (2022) had studied geometrical properties of the di�erent class of functions. In the present paper, we introduce a new generalized class of tilted analytic univalent functions S∗(θ,α,s,t) as the class of normalized functions f ∈ U ful�lling the condition Re ( eiθ zf ′(z) m(z) ) > α, (z ∈∆) for cos θ > α, |θ| < π, 0 ≤ α <1, m(z)= z (1−sz)(1−tz), s, 1, -1 ≤ t < 1, s , t and s, t ∈ C. Thus, our purpose in the present paper is to obtain representation theorem and coe�cient bound of S∗(θ,α,s,t)f. https://crossmark.crossref.org/dialog/?doi=10.26554/sti.2022.7.1.67-72&domain=pdf https://doi.org/10.26554/sti.2022.7.1.67-72 Yahya et. al. Science and Technology Indonesia, 7 (2022) 67-72 2. REPRESENTATION THEOREM Suppose that P is the class of all functions with a positive real part ∆={w:|w|<1}. Thus, we can write the function p(z) ∈ P= p(z)= 1+ p1z+ p2z2+...+ pnzn+... as that is analytic in ∆={w : |w| < 1} such that Re {p(z)}>0. Next, we relate the func- tions S∗(θ,α,s,t) to the functions in P. For any f ∈ U, we have eiθ zf ′(z) m(z) −α − isinθ cosθ −α = P, (z, ∈,∆) (2) The approach of The Herglotz Representation Theorem will be carried on for the establishment of the representation theorem S∗(θ,α,s,t) 2.1 Theorem Let f ∈ S∗(θ,α,s,t), then some probability measure µ on the unit circle X f can be shown as f (z)= ∫ x [[ log(1−xsz) x − log(1−xtz) x ] + ( 2Aθαe −1θ ) [ − log(1−xz) (s−1)− (t−1)x + log(1−xsz) s (t− s)− (s−1)x − t log(1−xtz) (t− s)− (t−1)x ] ( 2αeiθ−e −2iθ (t− s) )] dµ(x) (3) and conversely, i f t is given by the (3), then f ∈ S∗(θ,α,s,t). Proof. We have p ∈ P which is p(z)= ∫ x (1+xz)(1-xz)−1 dµ(x), |x|=1, for some probability measure µ on the unit circle X. Rearrang- ing yields, eiθ zf ′(z) m(z) z = p(z)(cosθ −α)+α + isinθ , and by replacing cosθ-α= Aθα that is always positive, we have f ′(z)= m(z) zeiθ [Aθαp(z)+α + isinθ] Therefore, f ′(z)= e−iθ m(z) z [Aθαp(z)+α + isinθ] (4) Substituting m(z)= z (1−sz)(1−tz) into (4), we have f ′(z)= e−iθ ( z (1− sz)(1− tz) ) ( 1 z ) [Aθαp(z)+α + isinθ] and f ′(z)= e−iθ ( 1 (1− sz)(1− tz) ) [Aθαp(z)+α + isinθ] (5) From, we have f ′(z)= e−iθ ( 1 (1− sz)(1− tz) ) Aθα ∫ x (1+xz) (1−xz) dµ(x)+α + isinθ  Then, we obtain f ′(z)= ∫ x e−iθ [ (α + isinθ)(1−xz)+ (1+xz)Aθα (1−xsz)(1−xtz)(1−xz) ] dµ(x) It follows that f (z)= z∫ 0 ∫ x e−iθ [ Aθα(1+xϕ)+ (α + isinθ)(1−xϕ) (1−xsϕ)(1−xtϕ)(1−xϕ) ] dµ(x)dϕ and f (z)= z∫ 0 ∫ x e−iθ [ (cosθ +xϕcosθ −α −xϕα + isinθ −xϕisinθ +α −xϕα) (1−xsϕ)(1−xtϕ)(1−xϕ) ] dµ(x)dϕ Then, f (z)= z∫ 0 ∫ x e−iθ [ (cosθ + isinθ)+xϕ(cosθ − isinθ)−2xϕ+α −xϕα) (1−xsϕ)(1−xtϕ)(1−xϕ) ] dµ(x)dϕ Since cos a+ i sin a= eia and cos a- i sin a= e−ia f (z)= z∫ 0 ∫ x [ 1+xϕ [ (e−2iθ cosθ)−2α(e−iθ) ] (1−xsϕ)(1−xtϕ)(1−xϕ) ] dµ(x)dϕ (6) Upon simpli�cation (6) , we have f (z)= z∫ 0 ∫ x [ (e−iθ −2α)e−iθ +1 (1−xsϕ)(1−xtϕ)(1−xϕ) −(e−iθ −2α)e−iθ (1−xsϕ)(1−xtϕ) ] dµ(x)dϕ By changing the integration order, we obtain that f (z)= ∫ x  z∫ 0 [ (−e−2iθ +2αe−iθ) (t − s) ( − s (1−xsϕ) + t (1−xtϕ) ) +(2Aθαe −iθ ) ( 1 (s−1)(t −1)(1− xϕ) − s2 (t− s)(s−1)(1−xsϕ) + t2 (t− s)(t−1)(1−xtϕ) )] dϕ ] dµ(x) (7) Integrating (7) φ concerning, © 2022 The Authors. Page 68 of 72 Yahya et. al. Science and Technology Indonesia, 7 (2022) 67-72 f (z)= ∫ x [[ log(1−xsz) (x) − log(1−xtz) (x) ] +(2Aθαe −iθ ) [ − log(1−xz) (s−1)(t−1)x + s log(1−xsz) (−s+ t)(s−1)x − t log(1−xtz) (−s+ t)(t−1)x ] ( 2αe−iθ − e−2iθ (−s+ t) )] dµ(x) which |x|= 1, and this is desired representation. 2.2 Corollary The extreme points S∗(θ, α, s, t) are the unit points masses fx(z)= ∫ x [[ log(1−xsz) (x) − log(1−xtz) (x) ] + (2Aθαe −iθ ) [ − log(1−xz) (−1+ s)(−1+ t)x + s log(1−xsz) (−s+ t)(−1+ s)x − t log(1−xtz) (−s+ t)(−1+ t)x ] ( 2αe−iθ − e−2iθ (−s+ t) )] , with |x|=1. Meanwhile, f ′x (z)= (−e−2iθ +2αe−iθ) (t − s) ( − s (1−xsz) + t (1−xtz) ) + (2Aθαe −iθ ) ( 1 (−1+ s)(−1+ t)(1− xz) − s2 (−s+ t)(−1+ s)(1−xsz) + t2 (−s+ t)(−1+ t)(1−xtz) ) , is the derivatives of the extreme points for S∗(θ, α, s, t) with |(x)|=1. Thus f is normalized functions since f (0)= ∫ x [[ log(1−xs(0)) (x) − log(1−xt(0)) (x) ] + (2Aθαe −iθ ) [ − log(1−x(0)) (−1+ s)(−1+ t)x + s log(1−xs(0)) (−s+ t)(−1+ s)x − t log(1−xt(0)) (−s+ t)(−1+ t)x ] ( 2αe−iθ − e−2iθ (−s+ t) )] = 0 and f ′(0)= (−e−2iθ + (2α)e−iθ) (−s+ t) ( t (1−xt(0)) − s (1−xs(0)) ) + (2Aθαe −iθ ) ( 1 (s−1)(t −1)(1− x(0)) − s2 (t− s)(s−1)(1−xs(0)) + t2 (t− s)(t−1)(1−xt(0)) ) = 1 3. MAIN RESULT Now, we proceed on �nding coe�cient bound of S∗(θ, α, s, t)f. 3.1 Theorem if f � S∗(θ, α, s, t), then |an | 5 s.sn + Aθα(2sn −4s.sn) n(−s+ t)(−s+1) + Aθα(2tn −4t.tn)− tn + t.tn n(−s+1)(−1+ t) + 2Aθα n(−1+ s)(−1+ t) for n=2,3,4,...and equality is attained for each n when f is an extreme point of S∗(θ, α, s, t). Proof. From (7), we have f (z)= z∫ 0  ∫ x [ − (−e−2iθ + e−iθ(2α))s (t − s)(1− xsϕ) − (e−iθ(2Aθα))s 2 (t − s)(s−1)(1− xsϕ) + (−e−iθ + (2α))te−iθ (−s+ t)(1− xtϕ) − e−iθ(2Aθα)t 2 (−s+ t)(−1+ t)(1− xtϕ) + 2e−iθ(Aθα) (−1+ s)(−1+ t)(1− xϕ) ] dµ(x) ] dϕ and f (z)= z∫ 0  ∫ x [( − (−e−2iθ + e−iθ(2α))s (−s+ t) − (e−iθ(2Aθα))s 2 (−s + t)(−1+ s) ) 1 (1− xsϕ) + ( (e−iθ(−e1θ +2α)t (−s+ t)(1− xtϕ) − e−iθ(2Aθα)t 2 (−s+ t)(−1+ t) ) 1 (1− xtϕ) + ( e−iθ(2Aθα) (−1+ s)(−1+ t) ) 1 (1− xϕ) ] dµ(x) ] dϕ (8) Next, (8) be written as f (z)= z∫ 0 [( − (−e−2iθ +2αe−iθ)s (−s+ t) − (2Aθαe −iθ)s2 (−s + t)(−1+ s) ) ∫ x ∞∑ 0 [ (sx)n ] dµ(x)(ϕ)n + ( (−e−2iθ +2αe−iθ)t (−s + t) − (2Aθαe −iθ)t2 (−s+ t)(−1+ t) ) ∫ x ∞∑ 0 [ (tx)n ] dµ(x)(ϕ)n + ( (2Aθαe −iθ) (−1+ s)(−1+ t) ) ∫ x ∞∑ 0 [ xndµ ] (x)(ϕ)n  dϕ (9) From (9), substituting n=0, then © 2022 The Authors. Page 69 of 72 Yahya et. al. Science and Technology Indonesia, 7 (2022) 67-72 f (z)=− (−e−iθ +2α)e−iθ(−1+ s)(−1+ t)s (−s+ t)(−1+ s)(−1+ t) − e−iθ(2Aθα)(−1+ t)s2 (−s+ t)(−1+ s)(−1+ t) + (−e−2iθ +2αe−iθ)(−1+ s)(−1+ t)t (−s+ t)(−1+ s)(−1+ t) − (2Aθαe−iθ)(−1+ s)t2 (−s+ t)(−1+ s)(−1+ s) + (2Aθαe−iθ)(−s+ t) (−s+ t)(−1+ s)(−1+ t) and f (z)= 1 (t − s)(s−1)(t −1) (e−2iθs2t −2αs2t −(e−2iθ)s2 − e−2iθ(st)+ (2αs2)e−iθ + (2αst)e−iθ −2αse−iθ + se−2iθ)− (2s2t −2s2)Aθαe −iθ + (st2(−e−2iθ) +2αe−iθst2 + e−2iθst + e−2iθt2 − e−iθ(2αst) − e−iθ(2αt2)− e−2iθt + e−iθ(2αt)+ (2st2 −2t2) Aθαe −iθ + (2t −2s)Aθαe −iθ ) (10) Upon simpli�cation the equation (10), we have f (z)= 1 (t − s)(s−1)(t −1) [ (s2t − s2 + s− st2 + t2 − t)e−2iθ +(−2s2t −2s+2s2 +2st2 −2t2 +2t)αe−iθ +(−2s2t −2s+2s2 +2st2 −2t2 +2t)αe−iθ +(−2s2t +2s2 +2st2 +2t −2t2 −2s)Aθαe −iθ ] Substituting Aθα= cosθ-α and cosθ= eiθ−e−iθ 2 , thus f (z)= 1 (t − s)(s−1)(t −1) [ (s2t − s2 + s− st2 + t2 − t)e−2iθ +(−2s2t +2s2 −2s+2st2 −2t2 +2t)αe−iθ +(−2s2t +2s2 +2st2 −2t2 +2t −2s)( 1+2e−iθ −2αe−iθ 2 )] and f (z)= 1 (t − s)(s−1)(t −1) + [ (−2s2t +2s2 −2s+2st2 −2t2 +2t)αe−iθ + (−s2t + s2 − s+ st2 − t2 + t)e−2iθ +(2s2t −2s2 +2s−2st2 +2t2 −2t)+ (−s2t + s2 −s+ st2 − t2 + t) ] = 1 (−s+ t)(s−1)(t −1) [(−s+ t)(s−1)(t −1)] = 1 Rewrite the equation (9) will yield to f (z)= w∫ 0 [ 1− ( (−e−2iθ +2αe−iθ)s (t − s) − (2Aθαe−iθ)s2 (t − s)(s−1) ) ∫ x ∞∑ 0 [(sx)n]dµ(x)(ϕ)n + ( (−e−2iθ +2αe−iθ)t (t − s) − (2Aθαe−iθ)t2 (t − s)(t −1) ) ∫ x ∞∑ 0 [(tx)n]dµ(x)(ϕ)n + ( (2Aθαe−iθ) (s−1)(t −1) ) ∫ x ∞∑ 0 [xndµ] (x)(ϕ)n  dϕ Integrating φ concerning gives us, f (z)=z− ( (−e−2iθ +2αe−iθ)s (t − s) − (2Aθαe−iθ)s2 (t − s)(s−1) ) ∫ x ∞∑ 2 [ (sx)n−1 n ] dµ(x)zn + ( (−e−2iθ +2αe−iθ)t (t − s) − (2Aθαe−iθ)t2 (t − s)(t −1) ) ∫ x ∞∑ 2 [ (tx)n−1 n ] dµ(x)zn + ( (2Aθαe−iθ) (s−1)(t −1) ) ∫ x ∞∑ 2 [ (x)n−1 n ] dµ(x)zn As a comparison above equation (1) , we have an =− ( e−iθ(−e−iθ +2α)s (−s+ t)n − e−iθ(2Aθα)s2 (−s+ t)(s−1)n ) ∫ x (sx)n nsx dµ(x)+ ( e−iθ(−e−iθ +2α)t (−s+ t)n − e−iθ(2Aθα)t2 (−s+ t)(t −1)n ) ∫ x (tx)n−1dµ(x)+ ( e−iθ(2Aθα) n(s−1)(t −1) ) ∫ x (x)n nx dµ(x) and |an | = ����[− (e−iθ(−e−iθ +2α)s(−s+ t)n − e−iθ(2Aθα)s2(−s+ t)(s−1)n ) (s)n−1 + ( e−iθ(−e−iθ +2α)t (−s+ t)n − e−iθ(2Aθα)t2 (−s+ t)(t −1)n ) (t)n−1 + ( e−iθ(2Aθα) n(s−1)(t −1) )]  ∫ x (x)n−1dµ(x)  ������ . Upon simpli�cation −e−2iθ + 2αe−iθ = 1-2 Aθαe−iθ , we have © 2022 The Authors. Page 70 of 72 Yahya et. al. Science and Technology Indonesia, 7 (2022) 67-72 |an | = 1 n ������  ∫ x (x)n−1dµ(x)  [( (2Aθα)s 2e−iθ (−s+ t)(s−1) − (1− [ eiθ ] 2Aθα)s (−s+ t) ) (s)n−1 + ( (1−2e−iθAθα)t (−s+ t) − e−iθ(2Aθα)t 2 (−s+ t)(t −1) ) (t)n−1 + ( e−iθ(2Aθα) (s−1)(t −1) )]���� . and |an | = 1 n ������  ∫ x (x)n−1dµ(x)  [ − ( s.sn −4Aθαs.sne−iθ − sn +2snAθαe−iθ (−s + t)(−1+ s) ) + ( t.tn −4Aθαt.tne−iθ − tn +2tnAθαe−iθ (−s+ t)(−1+ t) ) + ( 2Aθαe−iθ (−1+ s)(−1+ t) )]���� Then, since |e−iθ |=1, |an | = 1 n ������  ∫ x (x)n−1dµ(x)  [ − ( s.sn − sn + (2sn −4s.sn)Aθαe −iθ (−s+ t)(−1+ s) ) + ( t.tn − tn + (2tn −4t.tn)Aθαe −iθ (−s+ t)(−1+ s) ) + ( 2Aθαe −iθ (−s+ t)(−1+ s) )]���� ≤ 1n ∫ x |(x)n−1|dµ(x) [ s.sn − sn + (2sn −4s.sn)Aθα (−s+ t)(−1+ s) + t.tn − tn + (2tn −4t.tn)Aθα (−s + t)(−1+ s) + 2Aθα (−s+ t)(−1+ s) ] = sn+1 − sn + (2sn −4sn+1)Aθα (−s+ t)(−1+ s)n + tn+1 − tn + (2tn −4tn+1)Aθα (−s+ t)(−1+ s)n + 2Aθα (−s+ t)(−1+ s)n as required. 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