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Let      F α  λ  \mathfrak {F}_\alpha ^\lambda       be the class of functions      f ( z ) = z + a 2 z 2 + ⋯  f(z) = z + {a_2}{z^2} + \cdots       which are regular in      E = { z / | z | &gt; 1 } E = \{ z/|z| &gt; 1\}       and satisfy  \[    Re ⁡  { e i α  ( 1 + z f ( z ) / f ′ ( z ) ) } &gt; λ  cos ⁡  α  \operatorname {Re} \{ {e^{i\alpha }}(1 + zf(z)/f’(z))\} &gt; \lambda \cos \alpha     \]  for some      α  , | α  | &gt; π  / 2 \alpha ,|\alpha | &gt; \pi /2      , and for some      λ  , 0 ≤  λ  &gt; 1 \lambda ,0 \leq \lambda &gt; 1      . The author finds a range on      α  \alpha       for which      f ( z ) f(z)       in      F α  λ  \mathfrak {F}_\alpha ^\lambda       is univalent in      E E      . In particular, the author improves upon the range on a for which      f ( z ) ∈  F α  0 f(z) \in \mathfrak {F}_\alpha ^0       is known to be univalent in      E E      . Also a corresponding result is obtained for those functions      f ( z ) f(z)       in      F α  λ  \mathfrak {F}_\alpha ^\lambda       for which      f ( 0 ) = 0 f(0) = 0      .

Regular functions 𝑓(𝑧) for which 𝑧𝑓^{’}(𝑧) is 𝛼-spiral-like