Fand contemplate a holomorphic function in U provided by the equation

Fand look at a holomorphic function in U given by the equation h(z) = q(z) + zq (z) , h (0) = q (0), – 1 + q(z) (10)when q is a univalent resolution in U which satisfies the fuzzy differential subordination:|h(z)| |q(z)| i.e., q(z) 1 + |q(z)| 1 + |h(z)|Fh(z), z U.(11)Take into consideration (u, v; z), a confluent (or Kummer) hypergeometric function offered by (five) and the Vedaprofen Technical Information operator M (z) given by (7). If 1 +z (u,v;z) (u,v;z)is analytic in U, andz (u,v;z) (u,v;z)1+ 1+ thenz (u,v;z) 1 + (u,v;z)z (u, v; z) |h(z)| i.e., 1 + 1 + |h(z)| (u, v; z)Fh ( z ),(12)zM (z) M(z)1+zM (z) M(z)zM (z) |q(z)| i.e., 1 + |q(z)| M(z)Fq(z), z U,and q is the fuzzy ideal dominant. Proof. Relation (7) is equivalent to z -1 M ( z ) =z(u, v; t)t -1 dt.(13)Dimethoate web Differentiating (13) and soon after brief calculation we get( – 1) M(z) + zM (z) = (u, v; z) z,that is equivalent to M(z) – 1 + We let z 1 + 2 u +1 z + . . . v zM (z) = p(z) = u M(z) z 1 + v +1 z + . . .(14)zM (z) = (u, v; z) z. M(z)(15)=1 + 2 u +1 z + . . . v1 + u +1 z + . . . v,(16)p(0) = 1. Making use of (16) in (15), we get M (z)[ – 1 + p(z)] = (u, v; z) z. Differentiating (17), we receive zp (z) z (u, v; z) zM (z) + = + 1. M(z) – 1 + p(z) (u, v; z) Making use of (16) in (18), we have p(z) + zp (z) z (u, v; z) = + 1. – 1 + p(z) (u, v; z) (19) (18) (17)Mathematics 2021, 9,6 ofUsing (19) in (12), we get p(z) +zp (z) -1+ p ( z ) zp (z) -1+ p ( z )1 + p(z) +|h(z)| , z U. 1 + |h(z)|(20)In an effort to get the desired relation, Lemma 1 are going to be applied. To apply the lemma, let the function : C2 U C, (r, s; z) = r + s , r, s C. -1+r (21)For r = p(z), s = zp (z), relation (21) becomes p(z), zp (z) = p(z) + Making use of (22) in (20), we have zp (z) , z U. – 1 + p(z) (22)|h(z)| |( p(z), zp (z))| , z U. 1 + |( p(z), zp (z))| 1 + |h(z)|Applying Lemma 1, for = – 1, = = 0, we receive p(z), zp (z) Working with (22) in (24), we get p(z) + zp (z) – 1 + p(z)F F(23)h(z), z U.(24)h(z), z U.(25)In accordance with Lemma 1, relation (25) implies p(z) Applying (16) in (26) we’ve zM (z) M(z)F Fq(z), z U.(26)q(z), z U.(27)Given that q satisfies the differential Equation (ten), q is definitely the fuzzy ideal dominant. Remark four. Making use of particular expressions for the fuzzy ideal dominant q, sufficient conditions for starlikeness on the operator M (z) given by (7) might be obtained. If in Theorem 1, function q(z) =1- z 1+ zis regarded as the following corollary is obtained.Corollary 1. For , C, 1, 0, let the fuzzy function F : C [0, 1] offered by (9) and take into consideration a holomorphic function in U offered by the equation h(z) =( – 1) 1 – z2 + 1 – 4z + z2 ( – 1)(1 + z)two + 1 – z1- z 1+ z,h(0) = q(0) = 1, where function q(z) = fuzzy differential subordinationis a univalent option in U, which satisfies the( – 1) 1 – z2 + 1 – 4z + z2 |1 – z | , |1 + z | + |1 – z | ( – 1)(1 + z)two + 1 – z2 + |( – 1)(1 – z2 ) + 1 – 4z + z2 |Mathematics 2021, 9,7 ofi.e., 1-z 1+zF( – 1) 1 – z2 + 1 – 4z + z2 ( – 1)(1 + z)2 + 1 – z, z U.Take into account (u, v; z) to be a confluent (or Kummer) hypergeometric operator given by (5), and also the operator M (z) offered by (7). If 1 + 1+z (u,v;z) (u,v;z)is analytic in U, andz (u,v;z) (u,v;z) z (u,v;z) (u,v;z)( – 1) 1 – z2 + 1 – 4z + z2 ( – 1)(1 + z)two + 1 – z2 + |( – 1)(1 – z2 ) + 1 – 4z + z2 |z (u, v; z) (u, v; z)1- z 1+ z1+ 1+i.e., 1 + then( – 1) 1 – z2 + 1 – 4z + zF( – 1)(1 + z)2 + 1 – zzM (z) M(z),zM (z) M(z)1+ and q(z) =1- z 1+ zzM (z) M(z)1+1- z 1+ zi.e.,F1-z , z U, or M S 1+zis the fuzzy best dominant.Proof. By using the function q(z) = 1-z in relation (27).

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