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This semester I got introduced to complex analysis. While starting my studies I came across this questiom. What is a good complex analysis textbook, barring Ahlfors's?. And I exactly followed top answer. I started with Churchill and for geometric intuition I always look in needham. I enjoy the process truly. Now I want to take things ahead. I want to go to some advance level. Mainly I am looking for a book in which there is good explanation about univalent functions as I have to present seminar next semester on this topic. Suggestions will be really helpful.

ogirkar
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    This may seem cliche, but if you want the advanced level then Ahlfors is quite decent. I personally really love Narasimham's Complex Analysis in One Variable, but that might be a bit advanced... it's actually comprehensive and starts with into complex analysis, but requires a very mathematically mature reader and moves to research level stuff quite quickly. I know some who have really enjoyed Conway's books, though some say they are a tad dry; his second book in particular might interest you though. Is there something about univalent functions you wish to study? – Brevan Ellefsen May 06 '19 at 05:43
  • There are also a number of directions you could go; for example, are you wanting to stay with a single complex variable in the plane, or do Riemann Surfaces (or more general manifolds) interest you? Are you wanting to classify some types of univalent functions or merely desire to study some of their properties? I can't say I am remotely an expert on univalent maps, and I am sure others here can you point you in a better direction than I, but perhaps I can help a bit if you expound a bit more on what exactly you want and what you have/haven't enjoyed studying thus far. – Brevan Ellefsen May 06 '19 at 05:44
  • Another possibility is Rudin's RCA, though I don't recall univalent functions being a big topic ever. Rudin is... an admittedly extremely hard author to parse, though I quite enjoyed Rudin's RCA even as an undergraduate studying complex analysis. Another question: what is your interest with complex analysis? Do you intend to pursue it? Is your field of study in Analysis? Topology? Number Theory? Each of these queries could perhaps help in tailoring a book recommendation – Brevan Ellefsen May 06 '19 at 05:53
  • @Brevan Ellefsen...now I would like to study more single complex variable in plane only..and about univalent functions as i want to get introduced to it first but at same time I also have to present one small seminar on it, I am looking for a book which have provided good introduction to it and have taken it to some good level( yes mainly I would like to study their properties) – ogirkar May 06 '19 at 08:19
  • @Brevan Ellefsen...and to be honest I am not super excellent kind of student( but I am above average)..so book which will provide smooth transition from basic to advanced level will do great to me. – ogirkar May 06 '19 at 08:23
  • hmm. If your goals are to study univalent functions, I would recommend reading papers and short articles online. I am unaware of a common book on general complex analysis that has more than a smidgen on univalent functions (this is not to say one does not exist and I am merely unaware of it). For example, a quick search yields this article. The complex analysis in that article is quite elementary; in order to gauge the level you are at, how much of that article can you read? The article is roughly Ahlfors difficulty on a glance – Brevan Ellefsen May 06 '19 at 08:47
  • Thank you Sir..I will go through that article. – ogirkar May 06 '19 at 16:13

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Conway's books I & II might interest you. Steven Krantz also has some more advanced books, such as this book or this book.

However, as mentioned in the comments, complex analysis can lead various directions in harmonic analysis, algebraic geometry, number theory, complex manifolds, several complex variables, Riemann surfaces, etc. So the advanced material that might be of interest to you may depend on which direction you want to take it.

gs.co
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