Now I'll try to not retort with BBC 5 comments 
If you'll play I'll have a crude attempt at actually making the point they struggle with.
Now the question you pose is why pay Rafa 4 mil when he does a worse job than GH..
I think this a little unfair the question is Why should he be given more time? which is easier to answer.
But lets first try, now if we take the Arsenal match team...
Reina
Magic Cara Agger Aurelio
Masch Aquilani
kuyt Gerrard Benayoun
Torres
Now with an exception of a couple of players some already there, the odd one i.e Agger vs Hypia surely that is an upgrade from GH's team? Then you have Ngog/Reira
Now some arguments are based on so many transfers but how many are reserves or youth? For sure he's made mistakes as has every manager. But the last 2 seasons he hasn't been given a huge amount of funds...
Glen Johnson £17,500,000 Sami Hyypiä Free
Chris Mavinga Signed Jack Hobbs Signed
Stephen Sama Free Paul Anderson £250,000
Alberto Aquilani £17,000,000 Sebastian Leto £1,300,000
Sotirios Kyrgiakos £1,500,000 Astrit Ajdarevic Free
Jermaine Pennant Free
Alvaro Arbeloa £3,500,000
Xabi Alonso £30,000,000
Shane O Connor Free
Adam Hammill Signed
Total £36,000,000 £35,050,000 £950,000
Philipp Degen Free Anthony Le Tallec Undisclosed
Andrea Dossena £7,000,000 Harry Kewell Free
Diego Cavalieri £3,500,000 Peter Crouch £10,000,000
Robbie Keane £19,000,000 Danny Gutherie £2,500,000
Vitor Flora Signed John Arne Riise £4,000,000
Alberto Riera £8,000,000 Scott Carson £3,250,000
David Ngog £1,500,000 Steve Finnan £1,000,000
Robbie Keane £12,000,000
Total £39,000,000 £32,750,000 £6,250,000
Now that is little over 7 Million is this enough for a team contending for 4th place?
Then if we factor wages in as well
Chelsea 149Mil
Manu 121Mil
Arsenal 101mil
Liverpool 89Mil
So recently he hasn't been given the funds and the wage bill doesn't match Arsenal. So it some where inbetween Arsenal and Manu doesn't have the funds to buy willy nally and doesn't have the wages to offer to prospects. So his mistakes stand out more.
Now there is arguments for he is over achieving in the recent seasons so why do you feel he isn't over acheiving? Now we touched youth earlier so lets expand on this a little now Rafa wasn't happy with the youth setup(For sure you can criticise him for this but hindsight and all that)so ... Now he has in place what he feels is needed shouldn't he be given time to see if it bears fruit?
OK so I see some arguments seem to be placed around the style now(Here I will struggle but lets try), now you highlight several points from the Wigan match and sticking to plan but unless I'm mistaken didn't it start 4-4-2 no lone striker there, and also change it to 4-5-1? You also mention the defensive midfield now I accept there is some truth in that, but really OK mistakes have been made but in the short term(Lucas vs a temp signing) is Aquilani defensive? But if I'm honest I'm not going to defend him or condemn him for points lost this season or gained last season. The fact you've lost your midfield general has an impact now we can assign blame but lets just call it a mutual leaving in the end. So should Rafa not be given time to find his new midfield general? OK so you don't like his style but has style won Arsenal anything, didn't Mourhino get sacked for "style" reasons? What I'm trying to highlight is with the means given and certainly taking into account the last couple of seasons on the transfer market what else can he do, great attacking pacy scorers cost more than your Liverpool records?
Then the crux of it all what ever way it is dressed Rafa has done more in Europe than GH ever did as this is the money maker should he not continue? As it looks like revenue is a struggling point for Liverpool at the moment, with the owners mismanagement. In to my final bit can you trust your owners to appoint the correct person? The last person rumoured was Jurgen "bloody" Klinsmen. Coming from a club that has seen Souness and BS disasters mixed with other temps some worse some better, I ask is the road you wish to take or perhaps shouldn't you keep Rafa if for no other reason than stability? If you change it will also mean a huge change right through the club structure, could the club take that?
Firstly thank you for actually being able to think and to bother actually writting your opinion. There are a few good points made, and I will try and explain my views, or to offer the side of the argument.
BEFORE I DO, REDMAN, PLEASE STOP SIMPLY WRITING ABOUT ME, AND CONCENTRATE ON LIVERPOOL, ITS GETTING SILLY NOW. AS IVE ALREADY SAID TO YOUR MENTOR DUDE, LOWER CENTENARY ROW 21 SEAT 53. EVERY HOME GAME. COME AND HAVE A CHAT IF YOU ACTUALLY GO TO THE MATCH.
Ok, first you mention the team against Arsenal, so ok, lets compare it to the team he inherited from GH.
Reina - Dudek
Johnson - Finnan
Cara- Cara
Agger - Hyypia
Aurelio - Riise
Smicer - Kuyt
Gerrard - Gerrard
Masch - Hamman
Yossi - Kewell
Lucas - Baros
Torres - Ciise
Now, obviously as GH normally played 442, Ive had to compare Lucas to Baros, and I honestly would prefer Baros up front with Torres than Lucas in midfeld. I also didnt include Aqaulani because he didnt start.
Reina is better than Dudek, although shot stopping are similar, Reina has better distribution and more authority
Johnson vs Finnan is a close one, Johnson is better attacking, Finnan is a better defender, so it depends on what you are after
Agger vs Hyypia, goes to Sami, no offence to Agger, but Hyypia wins hands down
Aurelio vs Riise, again hands down to Riise
Smicer vs Kuyt, Closer than you may think. Smicer always promised more than he gave and Rafa promises more than Kuyt gives! As attacking threats, I would have to give it to Smicer, as a team player it goes to Dirk, so overall, I suppose Dirk shades it
Masch vs Hamman. Again terribly close, they are actually very similar, so I would call this a draw.
Yossi vs Kewell, Hands down to Yossi, but then a traffic cone could win this one
Torres vs Ciise, Again hands down to Torres, obviously.
So thats 4 to Rafa, 2 to GH and a draw. So basically in 5 years and after £228m and 76 attempts RB has 2 better players in his team than GH.
I shall deal with the rest in a bit, need something to eat first